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THE SCIENCE OF MECHANICS 




PROFESSOR ERNST MACH 
L838- 1916 




THE 



// 



SCIENCE OF MECHANICS 






z 



A CRITICAL AND HISTORICAL ACCOUNT 
OF ITS DEVELOPMENT 



DR. ERNST MACH 

PROFESSOR OF THE HISTORY AND THEORY OF INDUCTIVE SCIENCE IN 
THE UNIVERSITY OF VIENNA 



TRANSLATED FROM THE GERMAN 

BY 

THOMAS J. McCORMACK 






FOURTH EDITION 




CHICAGO LONDON 

HE OPEN COURT PUBLISHING CO. 
1919 



Copyright 1893, 1902, 1919 



Tlifc Ot>EN Court Publishing Company 



TRANSLATOR'S PREFACE TO THE 
SECOND ENGLISH EDITION. 

Since the appearance of the first edition of the 
present translation of Mach's Mechanics * the views 
which Professor Mach has advanced on the philoso- 
phy of science have found wide and steadily increas- 
ing acceptance. Many fruitful and elucidative con- 
troversies have sprung from his discussions of the 
historical, logical, and psychological foundations of 
physical science, and in consideration of the great 
ideal success which his works have latterly met with 
in Continental Europe, the time seems ripe for a still 
wider dissemination of his views in English-speaking 
countries. The study of the history and theory of 
science is finding fuller and fuller recognition in our 
universities, and it is to be hoped that the present ex- 
emplary treatment of the simplest and most typical 
branch of physics will stimulate further progress in 
this direction. 

The text of the present edition, which contains 
the extensive additions made by the author to the 

* Die Mechanik in ihrer Entwickelung historisch-kritisck dargestellt. Von 
Dr. Ernst Mach, Professor an der Universitat zu Wien. Mit 257 Abbildungen. 
First German edition, 1883. Fourth German edition, 1901. First edition of 
the English translation, Chicago, The Open Court Publishing Co., 1S93. 



vi TRANSLATOR'S PREFACE. 

latest German editions, has been thoroughly revised 
by the translator. All errors, either of substance or 
typography, so far as they have come to the trans- 
lator's notice, have been removed, and in many cases 
the phraseology has been altered. The sub-title of 
the work has, in compliance with certain criticisms, 
also been changed, to accord more with the wording 
of the original title and to bring out the idea that the 
work treats of the principles of mechanics predomi- 
nantly under the aspect of their development {Entwicke- 
lung). To avoid confusion in the matter of references, 
the main title stands as in the first edition. 

The author's additions, which are considerable, 
have been relegated to the Appendix. This course 
has been deemed preferable to that of incorporating 
them in the text, first, because the numerous refer- 
ences in other works to the pages of the first edition 
thus hold good for the present edition also, and sec- 
ondly, because with few exceptions the additions are 
either supplementary in character, or in answer to 
criticisms. A list of the subjects treated in these ad- 
ditions is given in the Table of Contents, under the 
heading "Appendix" on page xix. 

Special reference, however, must be made to the 
additions referring to Hertz's Mechanics (pp. 548-555), 
and to the history of the development of Professor 
Mach's own philosophical and scientific views, notably 
to his criticisms of the concepts of mass, inertia, ab- 
solute motion, etc., on pp. 542-547, 555-574, and 579 



TRANSLATOR'S PREFACE. vii 

-583. The remarks here made will be found highly 
elucidative, while the references given to the rich lit- 
erature dealing with the history and philosophy of 
science will also be found helpful. 

As for the rest, the text of the present edition of 
the translation is the same as that of the first. It has 
had the sanction of the author and the advantage 
of revision by Mr. C. S. Peirce, well known for his 
studies both of analytical mechanics and of the his- 
tory and logic of physics. Mr. Peirce read the proofs 
of the first edition and rewrote Sec. 8 in the chapter 
on Units and Measures, where the original was in- 
applicable to the system commonly taught in this 
country. 

Thomas J. McCormack. 
La Salle, III., February, 1902. 



AUTHOR'S PREFACE TO THE TRANS- 
LATION. 

Having read the proofs of the present translation 
of my work, Die Mechanik in ihrer Entwickelung, I can 
testify that the publishers have supplied an excellent, 
accurate, and faithful rendering of it, as their previous 
translations of essays of mine gave me every reason to 
expect. My thanks are due to all concerned, and 
especially to Mr. McCormack, whose intelligent care 
in the conduct of the translation has led to the dis- 
covery of many errors, heretofore overlooked. I may, 
thus, confidently hope, that the rise and growth of the 
ideas of the great inquirers, which it was my task to 
portray, will appear to my new public in distinct and 
sharp outlines. E. Mach. 

Prague, April 8th, 1893. 



PREFACE TO THE THIRD EDITION. 

That the interest in the foundations of mechanics 
is still unimpaired, is shown by the works published 
since 1889 by Budde, P. and J. Friedlander, H. Hertz, 
P. Johannesson, K. Lasswitz, MacGregor, K. Pearson, 
J. Petzoldt, Rosenberger, E. Strauss, Vicaire, P. 
Volkmann, E. Wohlwill, and others, many of which 
are deserving of consideration, even though briefly. 

In Prof. Karl Pearson {Grammar of Science, Lon- 
don, 1892), I have become acquainted with an inquirer 
with whose epistemological views I am in accord at 
nearly all essential points, and who has always taken 
a frank and courageous stand against all pseudo- 
scientific tendencies in science. Mechanics appears 
at present to be entering on a new relationship to 
physics, as is noticeable particularly in the publica- 
tion of H. Hertz. The nascent transformation in our 
conception of forces acting at a distance will perhaps 
be influenced also by the interesting investigations of 
H. Seeliger (" Ueber das Newton'sche Gravitations- 
gesetz, " Sitzungsbericht der Miinchener Akademie, 1 896), 
who has shown the incompatibility of a rigorous inter- 
pretation of Newton's law with the assumption of an 
unlimited mass of the universe. 

Vienna, January, 1897. E. Mach. 



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PREFACE TO THE FIRST EDITION. 



ing these cases must ever remain the method at once 
the most effective and the most natural for laying this 
gist and kernel bare. Indeed, it is not too much to 
say that it is the only way in which a real comprehen- 
sion of the general upshot of mechanics is to be at- 
tained. 

I have framed my exposition of the subject agree- 
ably to these views. It is perhaps a little long, but, on 
the other hand, I trust that it is clear. I have not in 
every case been able to avoid the use of the abbrevi- 
ated and precise terminology of mathematics. To do 
so would have been to sacrifice matter to form ; for the 
language of everyday life has not yet grown to be suf- 
ficiently accurate for the purposes of so exact a science 
as mechanics. 

The elucidations which I here offer are, in part, 
substantially contained in my treatise, Die Geschichte 
and die Wurzel des Satzes von der Erhaltung der Arbeit 
(Prague, Calve, 1872). At a later date nearly the same 
views' were expressed by Kirchhoff {Vorlesungen tiber 
mathematische Physik: Mechanik, Leipsic, 1874) and by 
Helmholtz (Die Thatsachen in der Wahrnehmung, 
Berlin, 1879), and have since become commonplace 
enough. Still the matter, as I conceive it, does not 
seem to have been exhausted, and I cannot deem my 
exposition to be at all superfluous. 

In my fundamental conception of the nature of sci- 
ence as Economy of Thought, a view which I in- 
dicated both in the treatise above cited and in my 



PREFACE TO THE FIRST EDITION. xi 

pamphlet, Die Gesialten der Fliissigkeit (Prague, Calve, 
1872), and which I somewhat more extensively devel- 
oped in my academical memorial address, Die okono- 
mische Natur der physikalischen Forschung (Vienna, Ge- 
rold, 1882, I no longer stand alone. I have been 
much gratified to find closely allied ideas developed, 
in an original manner, by Dr. R. Avenarius (Philoso- 
phie ah Denken der Welt, gemass dem Princip des klein- 
sten Krafimaasses, Leipsic, Fues, 1876). Regard for 
the true endeavor of philosophy, that of guiding into 
one common stream the many rills of knowledge, will 
not be found wanting in my work, although it takes a 
determined stand against the encroachments of m e te - speculq 
physical methods. 

The questions here dealt with have occupied me 
since my earliest youth, when my interest for them was 
powerfully stimulated by the beautiful introductions of 
Lagrange to the chapters of his Analytic Mechanics, as 
well as by the lucid and lively tract of Jolly, Principien 
der Mechanik (Stuttgart, 1852). If Duehring's esti- 
mable work, Kritische Geschichte der Principien der Me- 
chanik (Berlin, 1873), did not particularly influence 
me, it was that at the time of its appearance, my ideas 
had been not only substantially worked out, but actually 
published. Nevertheless, the reader will, at least on 
the destructive side, find many points of agreement 
between Duhring's criticisms and those here expressed. 

The new apparatus for the illustration of the sub- 
ject, here figured and described, were designed entirely 



xii PREFACE TO THE FIRST EDITION. 

by me and constructed by Mr. F. Hajek, the mechani- 
cian of the physical institute under my control. 

In less immediate connection with the text stand 
the f ac-simile reproductions of old originals in my pos- 
session. The quaint and naive traits of the great in- 
quirers, which find in them their expression, have al- 
ways exerted upon me a refreshing influence in my 
studies, and I have desired that my readers should 
share this pleasure with me. 

E. Mach. 

Prague, May, 1883. 



PREFACE TO THE SECOND EDITION. 

In consequence of the kind reception which this 
book has met with, a very large edition has been ex- 
hausted in less than five years. This circumstance and 
the treatises that have since then appeared of E. Wohl- 
will, H. Streintz, L. Lange, J. Epstein, F. A. Muller, 
J. Popper, G. Helm, M. Planck, F. Poske, and others 
are evidence of the gratifying fact that at the present 
day questions relating to the theory of cognition are 
pursued with interest, which twenty years ago scarcely 
anybody noticed. 

As a thoroughgoing revision of my work did not 
yet seem to me to be expedient, I have restricted my- 
self, so far as the text is concerned, to the correction 
of typographical errors, and have referred to the works 
that have appeared since its original publication, as 
far as possible, in a few appendices. 

E. Mach. 
Prague, June, 1888. 



PREFACE TO THE FIRST EDITION. 

The present volume is not a treatise upon the ap- 
plication of the principles of mechanics. Its aim is 
to clear up ideas, expose the real significance of the 
matter, and get rid of metaphysical obscurities. The 
little mathematics it contains is merely secondary to 
this purpose. 

Mechanics will here be treated, not as a branch of 
mathematics, but as one of the physical sciences. If 
the reader's interest is in that side of the subject, if 
he is curious to know how the principles of mechanics 
have been ascertained, from what sources they take 
their origin, and how far they can be regarded as 
permanent acquisitions, he will find, I hope, in these 
pages some enlightenment. All this, the positive and 
physical essence of mechanics, which makes its chief 
and highest interest for a student of nature, is in ex- 
isting treatises completely buried and concealed be- 
neath a mass of technical considerations. 

The gist and kernel of mechanical ideas has in al- 
most every case grown up in the investigation of very 
simple and special cases of mechanical processes ; and 
the analysis of the history of the discussions concern- 



PREFACE TO THE FOURTH EDITION. 

The number of the friends of this work appears to 
have increased in the course of seventeen years, and 
the partial consideration which my expositions have 
received in the writ'ngs of Boltzmann, Foppl, Hertz, 
Love, Maggi, Pearson, and Slate, have awakened in 
me the hope that my work shall not have been in 
vain. Especial gratification has been afforded me by 
finding in J. B. Stallo {The Concepts of Modern Physics') 
another staunch ally in my attitude toward mechanics, 
and in W. K. Clifford {Lectures and Essays and The 
Common Sense of the Exact Sciences), a thinker of kin- 
dred aims and points of view. 

New books and criticisms touching on my discus- 
sions have received attention in special additions, 
which in some instances have assumed considerable 
proportions. Of these strictures, O. Holder's note on 
my criticism of the Archimedean deduction {Denken 
und Anschauung in der Geometric, p. 63, note 62) has 
been of special value, inasmuch as it afforded me the 
opportunity of establishing my view on still firmer 
foundations (see pages 512-517). I do not at all dis- 
pute that rigorous demonstrations are as possible in 
mechanics as in mathematics. But with respect to 



xv i PREFACE TO THE FOURTH EDITION. 

the Archimedean and certain other deductions, I am 
still of the opinion that my position is the correct 
one. 

Other slight corrections in my work may have 
been made necessary by detailed historical research, 
but upon the whole I am of the opinion that I have 
correctly portrayed the picture of the transformations 
through which mechanics has passed, and presumably 
will pass. The original text, from which the later in- 
sertions are quite distinct, could therefore remain as 
it first stood in the first edition. I also desire that no 
changes shall be made in it even if after my death a 
new edition should become necessary. 

E. Mach. 
Vienna, January, 1901. 



TABLE OF CONTENTS. 

PAGE 

Translator's Preface to the Second Edition v 

Author's Preface to the Translation viii 

Preface to the First Edition ix 

Preface to the Second Edition xiii 

Preface to the Third Edition xiv 

Preface to the Fourth Edition xv 

Table of Contents xvii 

Introduction i 

CHAPTER I. 

THE DEVELOPMENT OK THE PRINCIPLES OF STATICS. 

I. The Principle of the Lever 8 

II. The Principle of the Inclined Plane 24 

III. The Principle of the Composition of Forces . . . . 33 

IV. The Principle of Virtual Velocities 49 

V. Retrospect of the Development of Statics 77 

VI. The Principles of Statics in Their Application to Fluids 86 
VII. The Principles of Statics in Their Application to Gas- 
eous Bodies no 

CHAPTER II. 

THE DEVELOPMENT OF THE PRINCIPLES OF DYNAMICS. 

I. Galileo's Achievements 128 

II. The Achievements of Huygens 155 

" III. The Achievements of Newton 187 

IV. Discussion and Illustration of the Principle of Reaction 201 
V. Criticism of the Principle of Reaction and of the Con- 
cept of Mass 216 

- VI. Newton's Views of Time, Space, and Motion .... 222 



xviii THE SCIENCE OF MECHANICS. 

PAGE 

VII. Synoptical Critique of the Newtonian Enunciations . 238 
VIII. Retrospect of the Development of Dynamics .... 245 

CHAPTER III. 

THE EXTENDED APPLICATION OF THE PRINCIPLES OF 
MECHANICS AND THE DEDUCTIVE DEVELOP- 
MENT OF THE SCIENCE. 

I. Scope of the Newtonian Principles 256 

** II. The Formulae and Units of Mechanics 269 

III. The Laws of the Conservation of Momentum, of the 

Conservation of the Centre of Gravity, and of the 

Conservation of Areas 287 

IV. The Laws of Impact 35 

V. D'Alembert's Principle 331 

VI. The Principle of Vis Viva 343 

VII. The Principle of Least Constraint 350 

VIII. The Principle of Least Action 364 

IX. Hamilton's Principle 380 

X. Some Applications of the Principles of Mechanics to 

Hydrostatic and Hydrodynamic Questions .... 384 

CHAPTER IV. 

THE FORMAL DEVELOPMENT OF MECHANICS. 

I. The Isoperimetrical Problems 421 

II. Theological, Animistic, and Mystical Points of View in 

Mechanics 446 

III. Analytical Mechanics . 465 

IV. The Economy of Science 481 

CHAPTER V. 

THE RELATION OF MECHANICS TO OTHER DEPART- 
MENTS OF KNOWLEDGE. 

I. The Relations of Mechanics to Physics 495 

II. The Relations of Mechanics to Physiology 504 






TABLE OF COXTEXTS. xix 

PAGE 

Appendix 509 

I. The Science of Antiquity, 509. II. Mechanical Researches of 
the Gretks, 510. III.. and IV. The Archimedean Deduction of the 
Law of the Lever, 512, 514. V. Mode of Procedure of Stevinus, 515. 
VI. Ancient Notions of the Nature of the Air, 517. VII. Galileo's 
Predecessors, 520. VIII. Galileo on Falling Bodies, 522. IX. Gali- 
leo on the Law of Inertia, 523. X. Galileo on the Motion of Projec- 
tiles, 525. XI. Deduction of the Expression for Centrifugal Force 
(Hamilton's Hodograph), 527. XII. Descartes and Huygens on 
Gravitation, 528. XIII. Physical Achievements of Huygens, 530. 
XIV. Newton's Predecessors, 531. XV. The Explanations of Gravi- 
tation, 533 XVI. Mass and Quantity of Matter, 536. XVII. Gali- 
leo on Tides, 537. XVIII. Mach's Definition of Mass, 539. XIX. 
Mach on Physiological Time, 541 XX. Recent Discussions of the 
Law of Inertia and Absolute Motion, 542. XXI. Tlertz's System of 
Mechanics, 548. XXII. History of Mach's Views of Physical Sci- 
ence (Mass, Inertia, etc.), 555. XXIII. Descartes's Achievements 
in Physics, 574. XXIV. Minimum Principles. 575. XXV. Grass- 
mann's Mechanics, 577 XXVI. Concept of Cause. 579 XXVII. 
Mach's Theory of the Economy of Thought, 579. XXVIII. Descrip- 
tion of Phenomena by Differential Equations, 583 XXIX. Mayer 
and the Mechanical Theory of Heat, 584. XXX. Principle of En- 
ergy, 585. 

Chronological Table of a Few Eminent Inquirers and of Their 

More Important Mechanical Works 589 

Index 593 



THE SCIENCE OF MECHANICS 



INTRODUCTION. 



i. That branch of physics which is at once the old- The science 
est and the simplest and which is therefore treated ics. 
as introductory to other departments of this science, 
is concerned with the motions and equilibrium of 
masses. It bears the name of mechanics. 

2. The history of the development of mechanics, 
is quite indispensable to a full comprehension of the 
science in its present condition. It also affords a sim- 
ple and instructive example of the processes by which 
natural science generally is developed. 

An instinctive, irreflective knowledge of the processes instinctive 

knowledge. 

of nature will doubtless always precede the scientific, 
conscious apprehension, or investigation, of phenom- 
ena. The former is the outcome of the relation in 
which the processes of nature stand to the satisfac- 
tion of our wants. The acquisition of the most ele- 
mentary truth does not devolve upon the individual 
alone : it is pre-effected in the development of the race. 

In point of fact, it is necessary to make a dis- Mechanical 
tinction between mechanical experience and mechan- exper 
ical science, in the sense in which the latter term is at 
present employed. Mechanical experiences are, un- 
questionably, very old. If we carefully examine the 
ancient Egyptian and Assyrian monuments, we shall 
find there pictorial representations of many kinds of 



THE SCIENCE OF MECHANICS. 



The me- implements and mechanical contrivances ; but ac- 
knowledge counts of the scientific knowledge of these peoples 
of antiquity ^ e e i t h er totally lacking, or point conclusively to a 
very inferior grade of attainment. By the side of 

highly ingenious ap- 
pliances, we behold 
the crudest and rough- 
est expedients em- 
ployed as the use of 
sleds, for instance, for 
the transportation of 
enormous blocks of 
stone. All bears an 
instinctive, unperfec- 
ted, accidental char- 
acter. 

So, too, prehistoric 
graves contain imple- 
ments whose construc- 
tion and employment 
imply no little skill 
and much mechanical 
experience. Thus, long 
before theory was 
dreamed of, imple- 
ments, machines, me- 
chanical experien- 
ces, and mechanical 
knowledge were abun- 
dant. 

3. The idea often 
suggests itself that 
perhaps the incom- 
plete accounts we pos- 




INTRODUCTION. 3 

sess have led us to underrate the science of the ancient 
world. Passages occur in ancient authors which seem 
to indicate a profounder knowledge than we are wont 
to ascribe to those nations. Take, for instance, the 
following passage from Yitruvius, De Architectura, 
Lib. V, Cap. Ill, 6: 

"The voice is a flowing breath, made sensible to a passage 
"the organ of hearing by the movements it produces^ %Itni * 
"in the air. It is propagated in infinite numbers of 
"circular zones- exactly as when a stone is thrown 
"into a pool of standing water countless circular un- 
"dulations are generated therein, which, increasing 
"as they recede from the centre, spread out over a 
"great distance, unless the narrowness of the locality 
"or some obstacle prevent their reaching their ter- 
"mination ; for the first line of waves, when impeded 
"by obstructions, throw by their backward swell the 
"succeeding circular lines of waves into confusion. 
" Conformably to the very same law, the voice also 
" generates circular motions ; but with this distinction, 
" that in water the circles, remaining upon the surface, 
"are propagated horizontally only, while the voice is 
"propagated both horizontally and vertically." 

Does not this sound like the imperfect exposition controvert- 
of a popular author, drawn from more accurate disqui- evidence, 
sitions now lost? In what a strange light should we 
ourselves appear, centuries hence, if our popular lit- 
erature, which by reason of its quantity is less easily 
destructible, should alone outlive the productions of 
science ? This too favorable view, however, is very 
rudely shaken by the multitude of other passages con- 
taining such crude and patent errors as cannot be con- 
ceived to exist in any high stage of scientific culture. 
(See Appendix, I., p. 509.) 



4 THE SCIENCE OF MECHANICS. 

The origin 4. When, where, and in what manner the develop- 

of science 

ment of science actually began, is at this day difficult 
historically to determine. It appears reasonable to 
assume, however, that the instinctive gathering of ex- 
periential facts preceded the scientific classification of 
them. Traces of this process may still be detected in 
the science of to-day; indeed, they are to be met with, 
now and then, in ourselves. The experiments that 
man heedlessly and instinctively makes in his strug- 
gles to satisfy his wants, are just as thoughtlessly and 
unconsciously applied. Here, for instance, belong the 
primitive experiments concerning the application of 
the lever in all its manifold forms. But the things 
that are thus unthinkingly and instinctively discovered, 
can never appear as peculiar, can never strike us as 
surprising, and as a rule therefore will never supply an 
impetus to further thought. 

Thefunc- The transition from this stage to the classified, 

' cia" S casses scientific knowledge and apprehension of facts, first be- 

veiopment comes possible on the rise of special classes and pro- 
fessions who make the satisfaction of definite social 
wants their lifelong vocation. A class of this sort oc- 
cupies itself with particular kinds of natural processes. 
The individuals of the class change ; old members 
drop out, and new ones come in. Thus arises a need 
of imparting to those who are newly come in, the 
stock of experience and knowledge already possessed ; 
a need of acquainting them with the conditions of the 

The com- attainment of a definite end so that the result may be 

munication . 1 1 r 1 1 /tm r 1 1 

of knowi- determined beforehand. 1 he communication ot knowl- 
edge is thus the first occasion that compels distinct re- 
flection, as everybody can still observe in himself. 
Further, that which the old members of a guild me- 
chanically pursue, strikes a new member as unusual 



edge. 



INTRODUCTION. 5 

and strange, and thus an impulse is given to fresh re- 
flection and investigation. 

When we wish to bring to the knowledge of a per- involves 

description. 

son any phenomena or processes of nature, we have 
the choice of two methods : we may allow the person to 
observe matters for himself, when instruction comes 
to an end ; or, we may describe to him the phenomena 
in some way, so as to save him the trouble of per- 
sonally making anew each experiment. Description, 
however, is only possible of events that constantly re- 
cur, or of events that are made up of component 
parts that constantly recur. That only can be de- 
scribed, and conceptually represented which is uniform 
and conformable to law ; for description presupposes 
the employment of names by which to designate its 
elements ; and names can acquire meanings only when 
applied to elements that constantly reappear. 

5. In the infinite variety of nature many ordinary a unitary 

^ * J conception 

events occur; while others appear uncommon, per- of nature, 
plexing, astonishing, or even contradictory to the or- 
dinary run of things. As long as this is the case we 
do not possess a well-settled and unitary conception of 
nature. Thence is imposed the task of everywhere 
seeking out in the natural phenomena those elements 
that are the same, and that amid all multiplicity are 
ever present. By this means, on the one hand, the 
most economical and briefest description and com- 
munication are rendered possible ; and on the other, The nature 
when once a person has acquired the skill of recog-edge. 
nising these permanent elements throughout the great- 
est range and variety of phenomena, of seeing them in 
the same, this ability leads to a comprehensive, compact, 
consistent, and facile conception 0/ the facts. When once 
we have reached the point where we are everywhere 



6 THE SCIENCE OF MECHANICS. 

The adap- able to detect the same few simple elements, combin- 

tation of 

thoughts toing in the ordinary manner, then they appear to us as 

facts. 

things that are familiar ; we are no longer surprised, 
there is nothing new or strange to us in the phenom- 
ena, we feel at home with them, they no longer per- 
. plex us, they are explained. It is a process of adaptation 
of thoughts to facts with which we are here concerned. 
Theecon- 6. Economy of communication and of apprehen- 

hought. sion is of the very essence of science. Herein lies 
its pacificatory, its enlightening, its refining element. 
Herein, too, we possess an unerring guide to the his- 
torical origin of science. In the beginning, all economy 
had in immediate view the satisfaction simply of bodily 
wants. With the artisan, and still more so with the 
investigator, the concisest and simplest possible knowl- 
edge of a given province of natural phenomena a 
knowledge that is attained with the least intellectual 
expenditure naturally becomes in itself an econom- 
ical aim ; but though it was at first a means to an end, 
when the mental motives connected therewith are once 
developed and demand their satisfaction, all thought 
of its original purpose, the personal need, disappears. 
Further de- To find, then, what remains unaltered in the phe- 
of these nomena of nature, to discover the elements thereof 
and the mode of their interconnection and interdepend- 
ence this is the business of physical science. It en- 
deavors, by comprehensive and thorough description, 
to make the waiting for new experiences unnecessary ; 
it seeks to save us the trouble of experimentation, by 
making use, for example, of the known interdepend- 
ence of phenomena, according to which, if one kind of 
event occurs, we may be sure beforehand that a" certain 
other event will occur. Even in the description itself 
labor may be saved, by discovering methods of de- 



ideas. 



IXTRODUCTIOX. 7 

scribing the greatest possible number of different ob- Their pres- 

... . ,, . . ... ent discus- 

jects at once and in the concisest manner. All this will sion merely 

preparatory 

be made clearer by the examination of points of detail 
than can be done by a general discussion. It is fitting, 
however, to prepare the way, at this stage, for the 
most important points of outlook which in the course 
of our work we shall have occasion to occupy. 

7. We now propose to enter more minutely into the proposed 
subject of our inquiries, and, at the same time, without Treatment, 
making the history of mechanics the chief topic of 
discussion, to consider its historical development so 
far as this is requisite to an understanding of the pres- 
ent state of mechanical science, and so far as it does 
not conflict with the unity of treatment of our main 
subject. Apart from the consideration that we cannot 
afford to neglect the great incentives that it is in our 
power to derive from the foremost intellects of allTheincen- 

t'ves de- 

epochs, incentives which taken as a whole are more rived from 
fruitful than the greatest men of the present day are with the 
able to offer, there is no grander, no more intellectually lects of the 
elevating spectacle than that of the utterances of the 
fundamental investigators in their gigantic power. 
Possessed as yet of no methods, for these were first 
created by their labors, and are only rendered compre- 
hensible to us by their performances, they grapple with 
and subjugate the object of their inquiry, and imprint 
upon it the forms of conceptual thought. They that 
know the entire course of the development of science, 
will, as a matter of course, judge more freely and And the in- 

crcusc of 

more correctly of the significance of any present scien- power 

.-r , , ,,..,.... which such 

tine movement than thev, who limited in their views a contact 

, . . . ,. , lends. 

to the age in which their own lives have been spent, 
contemplate merely the momentary trend that the course 
of intellectual events takes at the present moment. 



CHAPTER I. 

THE DEVELOPMENT OF THE PRINCIPLES OF 

STATICS. 

i. 

THE PRINCIPLE OF THE LEVER. 

Theeariiest i. The earliest investigations concerning mechan- 

mechanical . , . . . . . . . 

researches ics of which we nave any account, the investigations 

related to . _ , . . . . . 

statics. Of the ancient Greeks, related to statics, or to the doc- 
trine of equilibrium. Likewise, when after the taking 
of Constantinople by the Turks in 1453 a fresh impulse 
was imparted to the thought of the Occident by the an- 
cient writings that the fugitive Greeks brought with 
them, it was investigations in statics, principally evoked 
by the works of Archimedes, that occupied the fore- 
most investigators of the period. (See p. 510.) 

Archimedes 2. Archimedes of Syracuse (287-2 1 2 B. C. ) left 

of Syracuse ,,.,,. , r .. .... 

(287-212 b. behind him a number of writings, of which several 

c). 

have come down to us in complete form. We will 
first employ ourselves a moment with his treatise De 
JEquiponderantibus, which contains propositions re- 
specting the lever and the centre of gravity. 

In this treatise Archimedes starts from the follow- 
ing assumptions, which he regards as self-evident : 
Axiomatic a. Magnitudes of equal weight acting at equal 

assump- 

tions of Ar- distances (from their point of support) are in equi- 

chimedes. ... r ' ^ 

libnum. 



THE PRIXCIPLES OF STATICS. g 

b. Magnitudes of equal weight acting at une- Axiomatic 

... ,. , . , assump- 

qual distances (from their point of support) are tions of Ar- 

.,.,.,, , chimedes. 

not in equilibrium, but the one acting at the 
greater distance sinks. 
From these assumptions he deduces the following 
proposition : 

c. Commensurable magnitudes are in equilib- 
rium when they are inversely proportional to their 
distances (from the point of support). 

It would seem as if analysis could hardly go be- 
hind these assumptions. This is, however, when we 
carefully look into the matter, not the case. 

Imagine (Fig. 2) a bar, the weight of which is 
neglected. The bar rests on a fulcrum. At equal dis- 
tances from the fulcrum we ap- 
pend two equal weights. That 
the two weights, thus circum- 
stanced, are in equilibrium, is 
the assumption from which Archi- 
medes starts. We might suppose that this was self- Analysis of 
evident entirely apart from any experience, agreeably to median as- 

. . . . . - *_ t*- sumptions. 

the so-called principle of sufficient reason ; that in view 
of the symmetry of the entire arrangement there is no 
reason why rotation should occur in the one direction 
rather than in the other. But we forget, in this, that 
a great multitude of negative and positive experiences 
is implicitly contained in our assumption ; the negative, 
for instance, that dissimilar colors of the lever-arms, 
the position of the spectator, an occurrence in the vi- 
cinity, and the like, exercise no influence ; the positive, 
on the other hand, (as it appears in the second as- 
sumption,) that not only the weights but also their dis- 
tances from the supporting point are decisive factors 
in the disturbance of equilibrium, that they also are cir- 



^^ 



io THE SCIENCE OF MECHANICS. 

cumstances determinative of motion. By the aid of 
these experiences we do indeed perceive that rest (no 
motion) is the only motion which can be uniquely* de- 
termined, or denned, by the determinative conditions 
of the case."f 
character Now we are entitled to regard our knowledge of 

and value of . . . . 

the Archi- the decisive conditions of any phenomenon as sufficient 

medean re- . .... . . 

suits. only in the event that such conditions determine the 

phenomenon precisely and uniquely. Assuming the 
fact of experience referred to, that the weights and 
their distances alone are decisive, the first proposition 
of Archimedes really possesses a high degree of evi- 
dence and is eminently qualified to be made the foun- 
dation of further investigations. If the spectator place 
himself in the plane of symmetry of the arrangement 
in question, the first proposition manifests itself, more- 
over, as a highly imperative instinctive perception, a 
result determined by the symmetry of our own body. 
The pursuit of propositions of this character is, fur- 
thermore, an excellent means of accustoming ourselves 
in thought to the precision that nature reveals in her 
processes. 

Thegenerai 3. We will now reproduce in general outlines the 

of the lever train of thought by which Archimedes endeavors to re- 
reduced to, . , ....... , 

the simple duce the general proposition of the lever to the par- 

and panic- . , , , , r . . ,. 

uiarcase. ticular and apparently self-evident case. The two 
equal weights i suspended at a and b (Fig. 3) are, if 
the bar ab be free to rotate about its middle point c, in 
equilibrium. If the whole be suspended by a cord at 
c, the cord, leaving out of account the weight of the 

* So as to leave only a single possibility open. 

t If, for example, we were to assume that the weight at the right de- 
scended, then rotation in the opposite direction also would be determined by 
the spectator, whose person exerts no influence on the phenomenon, taking 
up his position on the opposite side. 



THE PRIXCIPLES OF STATICS. 



bar, will have to support the weight 2. The equal 
weights at the extremities of the bar supply accor- 
dingly the place of the double weight at the centre. 



a 







& 



The genera] 
proposition 
of the lever 
reduced to 
the simple 
and partic- 
ular case. 



1 m [5~S 1 



Fig. 3. Fig. 4. 

On a lever (Fig. 4), the arms of which are in the 
proportion of 1 to 2, weights are suspended in the pro- 
portion of 2 to 1. The weight 2 we imagine replaced 
by two weights 1, attached on either side at a distance 
1 from the point of suspension. Now again we have 
complete symmetry about the point of suspension, and 
consequently equilibrium. 

On the lever-arms 3 and 4 (Fig. 5) are suspended 
the weights 4 and 3. The lever-arm 3 is prolonged 
the distance 4, the arm 4 is prolonged the distance 3, 
and the weights 4 and 3 are replaced respectively by 



/ * w * \ 


fi rH r 1 " 1 ri 


- 




!j 6 i A i 6 6 d 




> 


i, A A 

_i L-J U 



Fig- 5 

4 and 3 pairs of symmetrically attached weights \, 
in the manner indicated in the figure. Now again we 
have perfect symmetry. The preceding reasoning, The gener- 
which we have here developed with specific figures, is 
easily generalised. 

4. It will be of interest to look at the manner in 
which Archimedes's mode of view, after the precedent 
of Stevinus, was modified by Galileo. 



THE SCIENCE OF MECHANICS. 



Galileo's 
mode of 
treatment. 



m 


n 




m 


11 










7.171 


211 



Fig. 6. 



Galileo imagines (Fig. 6) a heavy horizontal prism, 
homogeneous in material composition, suspended by 
its extremities from a homogeneous bar of the same 
length. The bar is provided at its middle point 

with a suspensory attach- 
ment. In this case equi- 
librium will obtain ; this 
we perceive at once. But 
in this case is contained 
every other case, which 
Galileo shows in the 
following manner. Let 
us suppose the whole 
length of the bar or the prism to be 7.{tn-\- n). Cut 
the prism in two, in such a manner that one portion 
shall have the length im and the other the length in. 
We can effect this without disturbing the equilibrium 
by previously fastening to the bar by threads, close to 
the point of proposed section, the inside extremities of 
the two portions. We may then remove all the threads, 
if the two portions of the prism be antecedently at- 
tached to the bar by their centres. Since the whole 
length of the bar is i(m -f- n), the length of each half 
is m -f- n. The distance of the point of suspension of 
the right-hand portion of the prism from the point of 
suspension of the bar is therefore m, and that of the 
left-hand portion n. The experience that we have 
here to deal with the weight, and not with the form, 
of the bodies, is easily made. It is thus manifest, that 
equilibrium will still subsist if any weight of the mag- 
nitude 2? be suspended at the distance n on the one 
side and any weight of the magnitude in be suspended 
at the distance in on the other. The instinctive elements 
of our perception of this phenomenon are even more 



THE PRIXCIPLES OF STATICS. 13 

prominently displayed in this form of the deduction 
than in that of Archimedes. 

We may discover, moreover, in this beautiful pre- 
sentation, a remnant of the ponderousness which was 
particularly characteristic of the investigators of an- 
tiquity. 

How a modern physicist conceived the same prob- Lagrange's 

presenta- 

lem, may be learned from the following presentation of tion. 
Lagrange. Lagrange says : Imagine a horizontal ho- 
mogeneous prism suspended at its centre. Let this 
prism (Fig. 7) be conceived divided into two prisms 
of the lengths im and 2. If now we consider the 
centres of gravity of these two parts, at which we may 
imagine weights to act proportional to im and in, the 



2n 



Fig. 7- 

two centres thus considered will have the distances n 
and m from the point of support. This concise dis- 
posal of the problem is only possible to the practised 
mathematical perception. 

5. The object that Archimedes and his successors object of 
sought to accomplish in the considerations we have here and his suc- 
presented, consists in the endeavor to reduce the more 
complicated case of the lever to the simpler and ap- 
parently self-evident case, to discern the simpler in the 
more complicated, or vice versa. In fact, we regard 
a phenomenon as explained, when we discover in it 
known simpler phenomena. 

But surprising as the achievement of Archimedes 
and his successors may at the first glance appear to 
us, doubts as to the correctness of it, on further reflec- 



t r 



14 THE SCIENCE OF MECHANICS. 

Critique of tion, nevertheless spring up. From the mere assump- 

their meth- . . . 

ods. tion 01 the equilibrium 01 equal weights at equal dis- 

tances is derived the inverse proportionality of weight 
and lever-arm! How is that possible? If we were 
unable philosophically and a priori to excogitate the 
simple fact of the dependence of equilibrium on weight 
and distance, but were obliged to go for that result to 
experience, in how much less a degree shall we be able, 
by speculative methods, to discover the form of this 
dependence, the proportionality ! 

Thestaticai As a matter of fact, the assumption that the equi- 

moment in- .... 

voivedin librium- disturbing effect of a weight P at the distance 

all their de- ... 

ductions. L from the axis of rotation is measured by the product 
P.L (the so-called statical moment), is more or less 
covertly or tacitly introduced by Archimedes and all 
his successors. For when Archimedes substitutes for 
a large weight a series of symmetrically arranged pairs 
of small weights, which weights extend beyond the point 
of support, he employs in this very act the doctrine of 
the centre of gravity in its more general form, which is 
itself nothing else than the doctrine of the lever in its 

mo re general form. (See Appendix, III., p. 512.) 

without it Without the assumption above mentioned of the im- 

demonstra- r , , ,-- i 

tion is im- port oi the product P.L, no one can prove (Fig. 8) 

that a bar, placed in 
any way on the ful- 
crum S, is supported, 
with the help of a 

1 string attached to its 

7 ^S centre of gravity and 

F,g 8 " carried over a pulley, 

by a weight equal to its own weight. But this is con- 
tained in the deductions of Archimedes, Stevinus, 
Galileo, and also in that of Lagrange. 



possible. 



THE PRLXCIPLES OF STATICS. 



5 



6. Huycens, indeed, reprehends this method, and Huygens's 

criticism. 

gives a different deduction, in which he believes he has 
avoided the error. If in .. 
the presentation of La- 
grange we imagine the 
two portions into which 
the prism is divided 
turned ninety degrees 
about two vertical axes 
passing through the cen- 
tres of gravity *,/" of the 
prism-portions (see Fig. 
ga), and it be shown 
that under these circum- 
stances equilibrium still D 
continues to subsist, we 
shall obtain the Huygenian 




and simplified, it is as follows. 



Fig. 9- 
deduction. Abridged 
In a rigid weightless 




Pig PL 



plane (Fig. 9) through the point S we draw a straight 
line, on which we cut off on the one side the length 1 



i6 



THE SCIENCE OF MECHANICS. 



His own de- and on the other the length 2, at A and B respectively. 

auction. . . . . , . . 

On the extremities, at right angles to this straight 
line, we place, with the centres as points of contact, the 
heavy, thin, homogeneous prisms CD and FF, of the 
lengths and weights 4 and 2. Drawing the straight 
line HSG (where AG = %AC) and, parallel to it, the 
line CF, and translating the prism-portion CG by par- 
allel displacement to FH, everything about the axis 
GH is symmetrical and equilibrium obtains. But 
equilibrium also obtains for the axis AB ; obtains con- 
sequently for every axis through S, and therefore also 
for that at right angles to AB : wherewith the new 
case of the lever is given. 
Apparently Apparently, nothing else is assumed here than that 

unimpeach- 

able. equal weights p,p (Fig. 10) in the same plane and at 

equal distances /,/ from an axis AA' (in this plane) 
equilibrate one another. If we place ourselves in the 
plane passing through AA' perpendicularly to /,/, say 

Y ,K 



t 

i 


A 


P 






M ' 




I o 




P 




A 1 




Fig. 10. 



Fig. 11. 



at the point M, and look now towards A and now 
towards A', we shall accord to this proposition the 
same evidentness as to the first Archimedean proposi- 
tion. The relation of things is, moreover, not altered if 
we institute with the weights parallel displacements 
with respect to the axis, as Huygens in fact does. 



THE PRI.XCIPLES OF STATICS. 17 

The error first arises in the inference : if equilib- Yet invoiv- 

. r 1 1 -i . ing in the 

num obtains for two axes of the plane, it also obtains final infer- 

. ence an er- 

for every ether axis passing through the point of inter- ror. 
section of the first two. This inference (if it is not to 
be regarded as a purely instinctive one) can be drawn 
only upon the condition that disturbant effects are as- 
cribed to the weights proportional to their distances 
from the axis. But in this is contained the very kernel 
of the doctrine of the lever and the centre of gravity. 
Let the heavy points of a plane be referred to a 
in of rectangular coordinates (Fig. 11). The co- 
ordinates of the centre of gravity of a system of masses 
m vi m" . . . having the coordinates xx' x?' . . . yy'y" 
are, as we know, 

Mathemat- 

"Smx 2my fcl djscus- 

8, = 1) 5S= S10D * 

V,// ' ' 2m ' Huygens's 

inference. 

If we turn the system through the angle or, the new co- 
ordinates of the masses will be 

.Vj = x cosa ys'ma, y x = vcosor + .r sxnnc 
and consequently the coordinates of the centre of 

gravity 

,. 2lm (x cosa v sina) 2mx 2my 

, = ^ ^ = cosa -=-. sina-^ 

= B, cosa 77 sina 
and, similarl)-, 

if x = ;;cosa + sir\a. 
We accordingly obtain the coordinates of the new 
centre of gravity, by simply transforming the coordi- 
nates of the first centre to the new axes. The centre 
of gravity remains therefore the self-same point. If 
we select the centre of gravity itself as origin, then 
2mx=2my=0. On turning the system of axes, this 
relation continues to subsist. If, accordingly, equi- 



1 8 THE SCIENCE OF MECHANICS. 

librium obtains for two axes of a plane that are per- 
pendicular to each other, it also obtains, and obtains 
then only, for every other axis through their point of 
intersection. Hence, if equilibrium obtains for any 
two axes of a plane, it will also obtain for every other 
axis of the plane that passes through the point of in- 
tersection of the two. / 
The infer- These conclusions, however, are not deducible if 

enceadmis- . . 

sibie only the coordinates of the centre of gravity are determined 

on one con- . 

dition. by some other, more general equation, say 

$ = mf{x) + m'f{x') + m " f{x") + . . . 
m -f- m' -\- m" -{-... 

The Huygenian mode of inference, therefore, is in- 
admissible, and contains the very same error that we 
remarked in the case of Archimedes, 
seif-decep- Archimedes's self-deception in this his endeavor to 
chimedes". reduce the complicated case of the lever to the case 
instinctively grasped, probably consisted in his uncon- 
scious employment of studies previously made on the 
centre of gravity by the help of the very proposition he 
sought to prove. It is characteristic, that he will not 
trust on his own authority, perhaps even on that of 
others, the easily presented observation of the import 
of the product P.L, but searches after a further verifi- 
cation of it. 

Now as a matter of fact we shall not, at least at 
this stage of our progress, attain to any comprehension 
whatever of the lever unless we directly discern in the < 
phenomena the product P.L as the factor decisive of 
the disturbance of equilibrium. In so far as Archi- j 
medes, in his Grecian mania for demonstration, strives 
to get around this, his deduction is defective. But re- I 
garding the import of P.L as given, the Archimedean I 



\ 



THE PRIXCIPLES OF STATICS. 



19 



deductions still retain considerable value, in so far as Function of 
the modes of conception of different cases are supported medean de- 
the one on the other, in so far as it is shown that one 
simple case contains all others, in so far as the same 
mode of conception is established for all cases. Im- 
agine (Fig. 12) a homogeneous prism, whose axis is 
AB, supported at its centre C. To give a graphical 
representation of the sum of the products of the weights 
and distances, the sum decisive of the disturbance of 
equilibrium, let us erect upon the elements of the axis, 
which are proportional to the elements of the weight, 
the distances as ordinates ; the ordinates to the right 









U p/ 








w 








T 






A 




c/ 


M 










J 
V 


I B 


X X 


R 











Fig. is. 

of C (as positive) being drawn upwards, and to the left illustration 
of C (as negative) downwards. The sum of the areas 
of the two triangles, A CD -+- CBE = 0, illustrates here 
the subsistence of equilibrium. If we divide the prism 
into two parts at M, we may substitute the rectangle 
MUWB for MTEB, and the rectangle MVXA for 
TMCAD, where TP=\TE and TR = \TD, and the 
prism-sections MB, MA are to be regarded as placed 
at right angles to AB by rotation about Q and S. 



THE SCIENCE OF MECHANICS. 



In the direction here indicated the Archimedean 
view certainly remained a serviceable one even after 
no one longer entertained any doubt of the significance 
of the product P.L, and after opinion on this point had 
been established historically and by abundant verifica- 
tion. (See Appendix, IV., p. 514.) 
Treatment 7. The manner in which the laws of the lever, as 

of the lever 

by modern handed down to us from Archimedes in their original 

physicists. . 

simple form, were further generalised and treated by 
modern physicists, is very interesting and instructive. 
Leonardo da Vinci (1452-1519), the famous painter 
and investigator, appears to have been the first to rec- 
ognise the importance of the general notion of the so- 




P w <2 

Fig. 13. 

Leonardo called statical moments. In the manuscripts he has 

Da Vinci . ..... 

1452 1519)- left us, several passages are found from which this 
clearly appears. He says, for example : We have a 
bar AD (Fig. 13) free to rotate about A, and suspended 
from the bar a weight P, and suspended from a string 
which passes over a pulley a second weight Q. What 
must be the ratio of the forces that equilibrium may ob- 1 
tain? The lever- arm for the weight P is not AD, but 
the "potential" lever AB. The lever-arm for the | 
weight Q is not AD, but the "potential" lever AC. 
The method by which Leonardo arrived at this view | 
is difficult to discover. But it is clear that he recog- 



i 



THE PRINCIPLES OF STATICS. 21 

nised the essential circumstances by which the effect 
of the weight is determined. 

Considerations similar to those of Leonardo daGuido 

y- tt Ubalcu 

Vinci are also found in the writings of Guido Ubaldi. 
8. We will now endeavor to obtain some idea of 
the way in which the notion of statical moment, by 
which as we know is understood the product of a force 
into the perpendicular let fall from the axis of rotation 
upon the line of direction of the force, could have been 
arrived at, although the way that really led to this 
idea is not now fully ascertainable. That equilibrium 
exists (Fig. 14) if we lay a 
cord, subjected at both sides 
to equal tensions, over a 
pulley, is perceived without 
difficulty. We shall always 
find a plane of symmetry for 
the apparatus the plane 
which stands at right angles FiR - x +- 

to the plane of the cord and bisects () the angle made 
by its two parts. The motion that might be supposed a method 
possible cannot in this case be precisely determined or the notion 

r r J m m of the stat- 

defined by anv rule whatsoever : no motion will there- >cai mo- 

ment might 

fore take place. If we note, now, further, that the mate- have been 

r . arrived at. 

rial of which the pulley is made is essential only to the 
extent of determining the form of motion of the points 
of application of the strings, we shall likewise readily 
perceive that almost any portion of the pulley may 
be removed without disturbing the equilibrium of 
the machine. Tne rigid radii that lead out to the tan- 
gential points of the string, are alone-essential. We 
see, thus, that the rigid radii (or the perpendiculars on 
the linear directions of the strings) play here a part 
similar to the lever-arms in the lever of Archimedes. 




THE SCIENCE OF MECHANICS. 



This notion 
derived 
from the 
considera- 
tion of a 
wheei and 
axle. 



Let us examine a so-called wheel and axle (Fig. 
15) of wheel-radius 2 and axle-radius 1, provided re- 
spectively with the cord-hung loads 1 and 2 ; an appa- 
ratus which corresponds in every respect to the lever 
of Archimedes. If now we place about the axle, in 
any manner we may choose, a second cord, which we 
subject at each side to the tension of a weight 2, the 
second cord will not disturb the equilibrium. It is 
plain, however, that we are also permitted to regard 





Fig. 15. Fig. 16. 

the two pulls marked in Fig. 16 as being in equilib- 
rium, by leaving the two others, as mutually destruc- 
tive, out of account. But we arrive in so doing, dis- 
missing from consideration all unessential features, at 
the perception that not only the pulls exerted by the 
weights but also the perpendiculars let fall from the 
axis on the lines of the pulls, are conditions deter- 
minative of motion. The decisive factors are, then, 
the products of the weights into the respective per- 
pendiculars let fall from the axis on the directions of 
the pulls ; in other words, the so-called statical mo- 
ments. 
The princi- 9. What we have so far considered, is the devel- 
lever ail- opment of our knowledge of the principle of the lever, 
explain the Quite independently of this was developed the knowl- 
chines. edge of the principle of the inclined plane. It is not 
necessary, however, for the comprehension of the ma- 



THE PRIXCIPLES OF STATICS. 



23 




Fig. 17. 



chines, to search after a new principle beyond that of 
the lever ; for the latter is sufficient by'itself. Galileo, 
for example, explains the inclined plane from the lever 
in the following manner. 
We have before us (Fig. 
17) an inclined plane, on 
which rests the weight 
Q, held in equilibrium 
by the weight P. Gali- 
leo, now, points out the 
fact, that it is not requisite that Q should lie directly 
upon the inclined plane, but that the essential point 
is rather the form, or character, of the motion 
of Q. We may, consequently, conceive the weight 
attached to the bar AC, perpendicular to the inclined 
plane, and rotatable about C. If then we institute a Galileo's 

explanation 

verv slight rotation about the point C, the weight will of the in- 
clined 
move in the element of an arc coincident with the in- plane by 

the lever. 

clined plane. That the path assumes a curve on the 
motion being continued is of no consequence here, 
since this further movement does not in the case of 
equilibrium take place, and the movement of the in- 
stant alone is decisive. Reverting, however, to the 
observation before mentioned of Leonardo da Vinci, 
we readily perceive the validity of the theorem Q. CB 
= P. CA or Q/P = CAjCB = ca/cb, and thus reach 
the law of equilibrium on the inclined plane. Once we 
have reached the principle of the lever, we may, then, 
easily apply that principle to the comprehension of 
the other machines. 




? 4 THE SCIENCE OF MECHANICS. 



THE PRINCIPLE OF THE INCLINED PLANE. 

stevinus i. Stevinus, or Stevin, (1548-1620) was the first 

(1548-1620) . 1 r , 

first investi- who investigated the mechanical properties of the m- 
mechanics clined plane ; and he did so in an eminently original 

of the in- . 

clined manner. If a weight lie (fr ig. 

plane. / . . , , , 

18) on a horizontal table, we 
perceive at once, since the 
pressure is directly perpendic- 
ular to the plane of the table, 
by the principle of symmetry, 
Fig - ,8, that equilibrium subsists. On a 

vertical wall, on the other hand, a weight is not at all 
obstructed in its motion of descent. The inclined plane 
accordingly will present an intermediate case between 
these two limiting suppositions. Equilibrium will not 
exist of itself, as it does on the horizontal support, but 
it will be maintained by a less weight than that neces- 
sary to preserve it on the vertical wall. The ascertain- 
ment of the statical law that obtains in this case, caused 
the earlier inquirers considerable difficulty. 
Hismodeof Stevinus's manner of procedure is in substance as 
i-eac mgi s n ows j^ e i ma gines a triangular prism with horizon- 
tally placed edges, a cross-section of which ABC is 
represented in Fig. 19. For the sake of illustration 
we will say that AB = iBC ; also that AC is horizon- 
tal. Over this prism Stevinus lays an endless string , 
on which 14 balls of equal weight are strung and tied i 
at equal distances apart. We can advantageously re- 
place this string by an endless uniform chain or cord. 
The chain will either be in equilibrium or it will not. 
If we assume the latter to be the case, the chain, sinc< 



THE PRINCIPLES OF STATICS. 



::" :=.-; :;- 



the conditions of the event are not altered by its mo- 
tion, must, when once actually in motion, continue to 
move for ever, that is, it must present a perpetual mo- 
tion, which Stevinus deems absurd. Consequently only 
the first case is conceivable. The chain remains in equi- fdbe 
librium. The symmetrical portion ADC may, there- --~i 
fore, without disturbing the equilibrium, be removed. 
The portion AB of the chain consequently balances 
the portion BC. Hence : on inclined planes of equal 
heights equal weights act in the inverse proportion of 
the lengths of the planes. 




H| :: 



In the cross-section of the prism in Fig. 20 let us 
imagine A C horizontal, BC vertical, and AB =^ zBC; 
furthermore, the chain- weights Q and P on AB and 
BC proportional to the lengths ; it will follow then that 



26 



THE SCIENCE OF MECHANICS. 



The as- 
sumptions 
of Stevi- 
nus's de- 
duction. 



Their in- 
stinctive 
character. 



Their cog- 
ency. 



Q/P=AB/BC=2. The generalisation is self-evi- 
dent. 

2. Unquestionably in the assumption from which 
Stevinus starts, that the endless chain does not move, 
there is contained primarily only a purely instinctive 
cognition. He feels at once, and we with him, that 
we have never observed anything like a motion of the 
kind referred to, that a thing of such a character does 
not exist. This conviction has so much logical cogency 
that we accept the conclusion drawn from it respecting 
the law of equilibrium on the inclined plane without the 
thought of an objection, although the law if presented 
as the simple result of experiment, or otherwise put, 
would appear dubious. We cannot be surprised at this 
when we reflect that all results of experiment are ob- 
scured by adventitious circumstances (as friction, etc.), 
and that every conjecture as to the conditions which are 
determinative in a given case is liable to error. That 
Stevinus ascribes to instinctive knowledge of this sort 
a higher authority than to simple, manifest, direct ob- 
servation might excite in us astonishment if we did not 
ourselves possess the same inclination. The questior 
accordingly forces itself upon us : Whence does this 
higher authority come ? If we remember that scientific 
demonstration, and scientific criticism generally cai 
only have sprung from the consciousness of the individ- 
ual fallibility of investigators, the explanation is not li 
to seek. We feel clearly, that we ourselves have con- 
tributed nothing to the creation of instinctive knowl- 
edge, that we have added to it nothing arbitrarily, but 
that it exists in absolute independence of our partici- 
pation. Our mistrust of our own subjective interpre- 
tation of the facts observed, is thus dissipated. 

Stevinus's deduction is one of the rarest fossil in- 



THE PRIXCIPLES OF STATICS. 27 

dications that we possess in the primitive history of Hiehhistor- 
mechanics, and throws a wonderful light on the pro- stevinus-s 

deduction. 

cess of the formation of science generally, on its rise 
from instinctive knowledge. We will recall to mind 
that Archimedes pursued exactly the same tendency 
as Stevinus, only with much less good fortune. In 
later times, also, instinctive knowledge is very fre- 
quently taken as the starting-point of investigations. 
Every experimenter can daily observe in his own per- 
son the guidance that instinctive knowledge furnishes 
him. If he succeeds in abstractly formulating what 
is contained in it, he will as a rule have made an im- 
portant advance in science. 

Stevinus's procedure is no error. If an error were The trust- 
contained in it, we should all share it. Indeed, it is of instinc- 
perfectly certain, that the union of the strongest in- edge, 
stinct with the greatest power of abstract formulation 
alone constitutes the great natural inquirer. This by 
no means compels us, however, to create a new mysti- 
cism out of the instinctive in science and to regard this 
factor as infallible. That it is not infallible, we very 
easily discover. Even instinctive knowledge of so 
great logical force as the principle of symmetry em- 
ployed by Archimedes, may lead us astray. Many of 
my readers will recall to mind, perhaps, the intellectual 
shock they experienced when they heard for the first 
time that a magnetic needle lying in the magnetic 
meridian is deflected in a definite direction away from 
the meridian by a wire conducting a current being car- 
ried along in a parallel direction above it. The instinc- 
tive is just as fallible as the distinctly conscious. Its only 
value is in provinces with which we are very familiar. 

Let us rather put to ourselves, in preference to 
pursuing mystical speculations on this subject, the 



28 THE SCIENCE OF MECHANICS. 

The origin question : How does instinctive knowledge originate 

of instinc- . . 

tiveknowi- and what are its contents? Everything which we ob- 

edge. . ... 

serve in nature imprints itself uncomprehended and un- 
analysed in our percepts and ideas, which, then, in their 
turn, mimic the processes of nature in their most gen- 
eral and most striking features. In these accumulated 
experiences we possess a treasure-store which is ever 
close at hand and of which only the smallest portion 
is embodied in clear articulate thought. The circum- 
stance that it is far easier to resort to these experi- 
, ences than it is to nature herself, and that they are, 
notwithstanding this, free, in the sense indicated, from 
all subjectivity, invests them with a high value. It 
is a peculiar property of instinctive knowledge that it 
is predominantly of a negative nature. We cannot so 
well say what must happen as we can what cannot hap- 
pen, since the latter alone stands in glaring contrast to 
the obscure mass of experience in us in which single 
characters are not distinguished, 
instinctive Still, great as the importance of instinctive knowl- 
and extern- edge may be, for discovery, we must not, from our 

al realities ... . . , .... 

mutually point ot view, rest content with the recognition of its 

condition . TTr . . . _ T , 

each other, authority. We must inquire, on the contrary : Under 
what conditions could the instinctive knowledge in 
question have originated? We then ordinarily find that 
the very principle to establish which we had recourse 
to instinctive knowledge, constitutes in its turn the fun- 
damental condition of the origin of that knowledge. 
And this is quite obvious and natural. Our instinctive 
knowledge leads us to the principle which explains that 
knowledge itself, and which is in its turn also corrobo- 
rated by the existence of that knowledge, which is a 
separate fact by itself. This we will find on close ex- 
amination is the state of things in Stevinus's case. 



THE PRIXCIPLES OF STATICS. 29 

x. The reasoning of Stevinus impresses us as soTneingen- 

, 1 1 1 , 1 uityofSte- 

highly ingenious because the result at which he arrives vinuss rea- 
apparently contains more than the assumption from 
which he starts. While on the one hand, to avoid con- 
tradictions, we are constrained to let the result pass, on 
the other an incentive remains which impels us to seek 
further insight. If Stevinus had distinctly set forth 
the entire fact in all its aspects, as Galileo subsequently 
did, his reasoning would no longer strike us as ingen- 
ious ; but we should have obtained a much more satis- 
factory and clear insight into the matter. In the 
endless chain which does not glide upon the prism, is 
contained, in fact, everything. We might say, the 
chain does not glide because no sinking of heavy bodies 
takes place here. This would not be accurate, how- 
. for when the chain moves many of its links really 
do descend, while others rise in their place. We must 
say, therefore, more accurately, the chain does not 
glide because for every bod v that could possiblv de- critique <a 

r Stevinus s 

scend an equally heavy body would have to ascend deduction, 
equally high, or a body of double the weight half the 
height, and so on. This fact was familiar to Stevinus, 
who presented it, indeed, in his theory of pulleys ; 
but he was plainly too distrustful of himself to lay 
down the law, without additional support, as also valid 
for the inclined plane. But if such a law did not exist 
universally, our instinctive knowledge respecting the 
endless chain could never have originated. With this 
our minds are completely enlightened. The fact that 
Stevinus did not go as far as this in his reasoning and 
rested content with bringing his (indirectly discovered) 
ideas into agreement with his instinctive thought, need 
not further disturb us. (See p. 515.) 

The service which Stevinus renders himself and his 



30 THE SCIENCE OF MECHANICS. 

The merit readers, consists, therefore, in the contrast and com- 

nus'sproce- parison of knowledge that is instinctive with knowledge 

that is clear, in the bringing the two into connection 

and accord with one another, and in the supporting 




Fig. 21. 

the one upon the other. The strengthening of mental 
view which Stevinus acquired by this procedure, we 
learn from the fact that a picture of the endless chain 
and the prism graces as vignette, with the inscription 
"Wonder en is gheen wonder," the title-page of his 



THE PKIXCIPLES OF STATICS. 



3i 



work Hypomnemata Mathematica (Leyden, 1605).* As 
a fact, every enlightening progress made in science is 
accompanied with a certain feeling of disillusionment. 
We discover that that which appeared wonderful to 
us is no more wonderful than other things which we 
know instinctively and regard as self-evident ; nay, 
that the contrary would be much more wonderful ; that 
everywhere the same fact expresses itself. Our puzzle 
turns out then to be a puzzle no more ; it vanishes into 
nothingness, and takes its place among the shadows 
of history. 

4. After he had arrived at the principle of the in- 
clined plane, it was easy for Stevinus to apply that 
principle to the other machines and to explain bv it 
their action. He makes, for example, the following 
application. 

We have, let us suppose, an inclined plane (Fig. 
22) and on it a load Q. We pass a string over the 
pulley A at the summit and imagine the load Q held in 
equilibrium by the load P. 
Stevinus, now, proceeds by 
a method similar to that 
later taken by Galileo. He 
remarks that it is not ne- 
cessary that the load Q 
should lie directly on the 
inclined plane. Provided 
only the form of the machine's motion be preserved, the 
proportion between force and load will in all cases re- 
main the same. We may therefore equally well conceive 
the load Q to be attached to a properly weighted string 
passing over a pulley D: which string is normal to the 

The title given is that of Willebrord Snell's Latin translation (1608) of 
Simon Stevin's Wisconstigt Gedackteniistn, Leyden, 1605. Trans. 



Enlighten- 
ment in 
science al- 
ways ac- 
companied 
withdisillu 
sionment. 



Explana- 
tion of the 
other ma- 
chines by 
Stevinus's 
principle. 




Fig. 22 



32 



THE SCIENCE OF MECHANICS. 



The funicu- 
lar machine 



And the 
special case 
of the paral- 
lelogram of 
forces. 



The general 
form of the 
last- men- 
tioned prin- 
ciple also 
employed. 



inclined plane. If we carry out this alteration, we 
shall have a so-called funicular machine. We now 
perceive that we can ascertain very easily the portion 
of weight with which the body on the inclined plane 
tends downwards. We have only to draw a vertical 
line and to cut off on it a portion ab corresponding to 
the load Q. Then drawing on aA the perpendicular 
be, we have PJQ = AC/AB = ac/ab. Therefore ac 
represents the tension of the string aA. Nothing pre- 
vents us, now, from making the two strings change 
functions and from imagining the load Q to lie on the 
dotted inclined plane EDF. Similarly, here, we ob- 
tain ad for the tension of the second string. In this 
manner, accordingly, Stevinus indirectly arrives at a 
knowledge of the statical relations of the funicular 
machine and of the so-called parallelogram of forces ; at 
first, of course, only for the particular case of strings 
(or forces) ac, ad at right angles to one another. 

Subsequently, indeed, Stevinus employs the prin- 
ciple of the composition and resolution of forces in 
a more general form ; yet the method by which he 

X 





Fig. 23 Fig. 24. 

reached the principle, is not very clear, or at least is 
not obvious. He remarks, for example, that if we 
have three strings AB, AC, AD, stretched at any 



THE PRIXCIPLES OF STATICS. 



33 




Fig. 25. 



given angles, and the weight P is suspended from the 
first, the tensions may be determined in the following 
manner. We produce (Fig. 23) AB to X and cut off 
on it a portion AE. Drawing from the point E, EF 
parallel to AD and EG paral- 
lel to AC, the tensions of AB, 
AC, AD are respectively pro- 
portional to AE, AF, AG. 

With the assistance of this 
principle of construction Ste- 
vinus solves highly compli- 
cated problems. He determines, for instance, the solution of 

, ... . ... other corn- 

tensions of a system of ramifying strings like that plicated 

illustrated in Fig. 24; in doing which of course he pr 

starts from the given tension of the vertical string. 

The relations of the tensions of a funicular polygon 
are likewise ascertained by construction, in the man- 
ner indicated in Fig. 25. 

We may therefore, by means of the principle of the General re- 
inclined plane, seek to elucidate the conditions of op- 
eration of the other simple machines, in a manner sim- 
ilar to that which we employed in the case of the prin- 
ciple of the lever. 



suit. 



THE PRINCIPLE OF THE COMPOSITION OF FORCES. 

i. The principle of the parallelogram of forces, at The princi- 
which Stevinus arrived and employed, (yet without ex- paraiieio- 
pressly formulating it,) consists, as we know, of the forces, 
following truth. If a body A (Fig. 26) is acted upon 
by two forces whose directions coincide with the lines 
AB and AC, and whose magnitudes are proportional to 
the lengths AB and AC, these two forces produce the 



34 



THE SCIENCE OF MECHANICS. 




Fig. 26. 



same effect as a single force, which acts in the direction 
of the diagonal AD of the parallelogram ABCD and is 
proportional to that diagonal. For instance, if on the 

strings AB, AC weights 
exactly proportional to the 
lengths AB, AC be sup- 
posed to act, a single 
weight acting on the string 
^4Z>exactly proportional to 
the length AD will produce the same effect as the first 
two. The forces AB and AC are called the compo- 
nents, the force AD the resultant. It is furthermore 
obvious, that conversely, a single force is replaceable 
by two or several other forces. 
Method by 2. We shall now endeavor, in connection with the 

which the 

general no- investigations of Stevinus, to give ourselves some idea 

tionofthe & 7 , , , , 

paraiieio- of the manner in which the 

?ram of fc Z U . . 

general proposition of the 
parallelogram of forces 
might have been arrived 
at. The relation, dis- 
covered by Stevinus, 
that exists between two 
mutually perpendicular 
forces and a third force 
that equilibrates them, we 
shall assume as (indi- 
rectly) given. We sup- 
pose now (Fig. 27) that 
there act on three strings 
OX, OY, OZ, pulls which 
Let us endeavor to determine the 
nature of these pulls. Each pull holds the two remain- 
ing ones in equilibrium. The pull OYwe will replace 




X\ 

Fig. 27. 

balance each other 



THE PRIXCIPLES OF ST A TICS. 35 

(following Stevinus's principle) by two new rectangular The dednc- 
pulls, one in the direction Ou (the prolongation of general 

* . pnnciple 

OX \, and one at right angles thereto in the direction from the 

-" . special case 

Ov. And let us similarly resolve the pull OZ in theofstevinus. 
directions Ou and Ow. The sum of the pulls in the di- 
rection Ou, then, must balance the pull OX, and the 
two pulls in the directions Ov and Ow must mutually 
destroy each other. Taking the two latter as equal 
and opposite, and representing them by Om and On, 
we determine coincidently with the operation the com- 
ponents Op and Oq parallel to Ou, as well also as the 
pulls Or, Os. Now the sum Op -f- Oq is equal and op- 
posite to the pull in the direction of OX ; and if we 
draw st parallel to O Y, or rt parallel to OZ, either line 
will cut off the portion Ot = Op + Oq : with which re- 
sult the general principle of the parallelogram of forces 
is reached. 

The general case of composition may be deduced a different 

. . mode of the 

in still another wav from the special composition of same de- 

. duction. 

rectangular forces. Let OA and OB be the two forces 
acting at O. For OB substitute 
a force OC acting parallel to ~^. 
OA and a force OD acting at 
right angles to OA. There 
then act for OA and OB the /T 
two forces OE = OA -f OC Fig * 

and OD, the resultant of which forces OF is at the same 
time the diagonal of the parallelogram OAFB con- 
structed on OA and OB as sides. 

3. The principle of the parallelogram of forces, The prin- 

when reached bv the method of Stevinus, presents it- presents u- 

.... , se 'f as an 

self as an indirect discovery. It is exhibited as a con- indirect 

discovery. 

sequence and as the condition of known facts. We 
perceive, however, merely that it does exist, not, as yet 




36 



THE SCIENCE OF MECHANICS. 



And is first 
clearly 
enunciated 
by Newton 
and Varig- 
non. 



The geo- 
metrical 
theorem 
employed 
by Varig- 
non. 



why it exists ; that is, we cannot reduce it (as in dy- 
namics) to still simpler propositions. In statics, in- 
deed, the principle was not fully admitted until the 
time of Varignon, when dynamics, which leads directly 
to the principle, was already so far advanced that its 
adoption therefrom presented no difficulties. The prin- 
ciple of the parallelogram of forces was first clearly 
enunciated by Newton in his Principles of Natural Phi- 
losop/iy. In the same year, Varignon, independently of 
Newton, also enunciated the principle, in a work sub- 
mitted to the Paris Academy (but not published un- 
til after its author's death), and made, by the aid of a 
geometrical theorem, extended practical application 
of it.* 

The geometrical theorem referred to is this. If we 
consider (Fig. 29) a parallelogram the sides of which 
are/ and q, and the diagonal is r, and from any point m 

in the plane of the par- 
allelogram we draw per- 
pendiculars on these 
three straight lines, 
which perpendiculars 
we will designate as 
u, v, w, then p . u -\- 
q . v = r . w. This is 
easily proved by draw- 
Fig. 29. Fig. 30. ing straight lines from m 
to the extremities of the diagonal and of the sides of 
the parallelogram, and considering the areas of the 
triangles thus formed, which are equal to the halves 
of the products specified. If the point ;// be taken 
within the parallelogram and perpendiculars then be 




* In the same year, 1687, Father Bernard Lami published a little appendix 
to his Traiti de mechanique, developing the same principle. Trans. 



THE PR IXC I PL ES OF STATICS. 37 

drawn, the theorem passes into the form p .u q . v 
1= /- . w. Finally, if m be taken on the diagonal and 
perpendiculars again be drawn, we shall get, since the 
perpendicular let fall on the diagonal is now zero, 
p . u q. v = or p . u = q.v. 

With the assistance of the observation that forces The deduc- 
are proportional to the motions produced by them in 
equal intervals of time, Varignon easily advances from 
the composition of motions to the composition of forces. 
Forces, which acting at a point are represented in 
magnitude and direction by the sides of a parallelo- 
gram, are replaceable by a single force, similarly rep- 
resented by the diagonal of that parallelogram. 

If now. in the parallelogram considered,/ and ^Momemsof 
represent the concurrent forces (the components) and r 
the force competent to take their place (the resultant), 
then the products pu, qv, ru> are called the moments 
of these forces with respect to the point m. If the point 
/// lie in the direction of the resultant, the two moments 
pu and qv are with respect to it equal to each other. 

4. With the assistance of this principle Varignon is varignon's 
now in a position to treat 
the machines in a much 
simpler manner than were 
his predecessors. Let us 
consider, for example, 
(Fig. 31) a rigid body 
capable of rotation about 
an axis passing through 
0. Perpendicular to the 
axis we conceive a plane, 
and select therein two F ' 8 ' 3I- 

points A, B, on which two forces /'and Q in the plane 
are supposed to act. W 7 e recognise with Varignon 




38 THE SCIENCE OF MECHANICS. 

The deduc-that the effect of the forces is not altered if their points 

tion of the f ...... . 

law of the of application be displaced along their line of action, 

lever from . ,. . , .. 

theparai- since all points in the same direction are rigidly con- 

lelogram- 

principle, nected with one another and each one presses and pulls 
the other. We may, accordingly, suppose P applied 
at any point in the direction AX, and Q at any point 
in the direction BY, consequently also at their point 
of intersection M. With the forces as displaced to M, 
then, we construct a parallelogram, and replace the 
forces by their resultant. We have now to do only 
with the effect of the latter. If it act only on movable 
points, equilibrium will not obtain. If, however, the 
direction of its action pass through the axis, through 
the point O, which is not movable, no motion can take 
place and equilibrium will obtain. In the latter case 
O is a point on the resultant, and if we drop the per- 
pendiculars u and v from O on the directions of the 
forces/, q, we shall have, in conformity with the the- 
orem before mentioned, p u = q v. With this we 
have deduced the law of the lever from the principle 
of the parallelogram of forces. 
The statics Varignon explains in like manner a number of other 
adynamicai cases of equilibrium by the equilibration of the result- 
ant force by some obstacle or restraint. On the in- 
clined plane, for example, equilibrium exists if the re- 
sultant is found to be at right angles to the plane. In 
fact, Varignon rests statics in its entirety on a dynamic 
foundation ; to his mind, it is but a special case of dy- 
namics. The more general dynamical case constantly 
hovers before him and he restricts himself in his inves- 
tigation voluntarily to the case of equilibrium. We 
are confronted here with a dynamical statics, such 
as was possible only after the researches of Galileo. 
Incidentally, it may be remarked, that from Varignon 



THE PKIXCIPLES OF STATICS. 



39 




is derived the majority of the theorems and methods 
of presentation which make up the statics of modern 
elementary text-books. 

5. As we have already seen, purely statical consid- 
erations also lead to the proposition of the parallel- 
ogram of forces. In special cases, in fact, the principle 
admits of being very easily verified. We recognise at 
once, for instance, that any number whatsoever of equal 
forces acting (by pull or pressure) in the same plane at 
a point, around which their suc- 
cessive lines make equal angles, 
are in equilibrium. If, for exam- 
ple, (Fig. 32) the three equal 
forces OA, OB, OC act on the 
point at angles of 120, each 
two of the forces holds the third 
in equilibrium. We see imme- 
diately that the resultant of OA 
and OB is equal and opposite to OC. It is represented 
by OD and is at the same time the diagonal of the 
parallelogram OADB, which readily follows from the 
fact that the radius of a circle is also the side of the 
hexagon included by it. 

6. If the concurrent forces act in the same or in 
opposite directions, the resultant, is equal to the sum 

or the difference of the 

n a 

components. We rec- 
ognise both cases with- 
out any difficulty as 
particular cases of the 
principle of the paral- 
lelogram of forces. If in the two drawings of Fig. 33 
we imagine the angle A OB to be gradually reduced 
to the value o', and the angle A' G B" increased to the 



Special 
statical con- 
siderations 
also lead to 
the prin- 
ciple. 



Fig. 3*- 




The case of 
coincident 
forces 
merely a 
particular 
cage of the 
general 
principle. 



Fig. 33- 



40 THE SCIENCE OF MECHANICS, 

value i8o, we shall perceive that OC passes into OA -(- 
AC= OA + OB And O' C into O' A' A' C = 0' A' 
0' B'. The principle of the parallelogram of forces 
includes, accordingly, propositions which are generally 
made to precede it as independent theorems. 
The princi- 7. The principle of the parallelogram of forces, in 

pea propo- 

sitionde- the form in which it was set forth by Newton and 

rived from . . j 

experience. Vangnon, clearly discloses itself as a proposition de- 
rived from experience. A point acted on by two forces 
describes with accelerations proportional to the forces 
two mutually independent motions. On this fact the 
parallelogram construction is based. Daniel Ber- 
noulli, however, was of opinion that the proposition of 
the parallelogram of forces was a geometrical truth, in- 
dependent of physical experience. And he attempted 
to furnish for it a geometrical demonstration, the chief 
features of which we shall here take into consideration, 
as the Bernoullian view has not, even at the present 
day, entirely disappeared. 

Daniel Ber- If two equal forces, at right angles to each other 

noulli's at- ,-,-.. . . . , , , 

tempted (r lg. 34), act on a point, there can be no doubt, ac- 

geomctrical .. _ ,,. , , ,. 

demonstra- n cording to Bernoulli, that the line 

tion of the a+ JL1 q . . . . r , . . 

truth. /|\ of bisection of the angle (con- 

formably to the principle of sym- 
metry) is the direction of the re- 
sultant r. To determine geomet- 
rically also the magnitude of the 
resultant, each of the forces / is 
Fl 8 34- decomposed into two equal forces 

q, parallel and perpendicular to r. The relation in 
respect of magnitude thus produced between / and q 
is consequently the same as that between r and p. We 
have, accordingly : 

p = )A . q and r = jx . /; whence r = ju 2 q. 












7' 










' 1 


9 ^ 




pS 




1 


V 



THE PRINCIPLES OF STATICS. 



Since, however, the forces q acting at right angles 
to r destroy each other, while those parallel to r con- 
stitute the resultant, it further follows that 

r = 2q; hence // = I 2, and r = Vl. p. 

The resultant, therefore, is represented also in re- 
spect of magnitude by the diagonal of the square con- 
structed on / as side. 

Similarly, the magnitude may be determined of the The case of 

unequal 

resultant of unequal rectangular components. Here, rectangular 

\ components 

however, nothing is known before- 
hand concerning the direction of 
the resultant r. If we decompose 
the components /, q (Fig. 35), ^ 
parallel and perpendicular to the 
yet undetermined direction r, into 
the forces u, s and v, t, the new 
forces will form with the compo- 
nents /, q the same angles that p, 

q form with r. From which fact the following relations 
in respect of magnitude are determined : 

p u q v q s p t 

from which two latter equations follows s = t =pqfr. 
On the other hand, however, 




=+ 



= / + *' 



The diagonal of the rectangle constructed on / and 
q represents accordingly the magnitude of the result- 
ant. 

Therefore, for all rhombs, the direction of the re- General re- 
sultant is determined ; for all rectangles, the magni- 
tude; and for squares both magnitude and direction. 
Bernoulli then solves the problem of substituting for 



42 THE SCIENCE OF MECHANICS. 

two equal forces acting at one given angle, other equal, 
equivalent forces acting at a different angle ; and finally 
arrives by circumstantial considerations, not wholly 
exempt from mathematical objections, but amended 
later by Poisson, at the general principle. 
Critique of 8. Let us now examine the physical aspect of this 

Bernoulli's . . . , . . , 

method. question. As a proposition derived from experience, 
the principle of the parallelogram of forces was already 
known to Bernoulli. What Bernoulli really does, there- 
fore, is to simulate towards himself a complete ignorance 
of the proposition and then attempt to philosophise 
it abstractly out of the fewest possible assumptions. 
Such work is by no means devoid of meaning and pur- 
pose. On the contrary, we discover by such proce- 
dures, how few and how imperceptible the experiences 
are that suffice to supply a principle. Only we must 
not deceive ourselves, as Bernoulli did ; we must keep 
before our minds all the assumptions, and should over- 
look no experience which we involuntarily employ. 
What are the assumptions, then, contained in Bernoul- 
li's deduction? 

The as- 9. Statics, primarily, is acquainted with force only 

sumptions ,, , , , , 

ofhisde- as a pull or a pressure, that from whatever source it 
rived from may come always admits of being replaced by the.pull 
experience. oj _ t ^ e p ressure { a weight. All forces thus may be re- 
garded as quantities of the same kind and be measured 
by weights. Experience further instructs us, that the 
particular factor of a force which is determinative of 
equilibrium or determinative of motion, is contained 
not only in the magnitude of the force but also in its 
direction, which is made known by the direction of the 
resulting motion, by the direction of a stretched cord, 
or in some like manner. We may ascribe magnitude 
indeed to other things given in physical experience, 




THE PRIXCIPLES OF STATICS. 43 

such as temperature, potential function, but not direc- 
tion. The fact that both magnitude and direction are 
determinative in the efficiency of a force impressed on 
a point is an important though it may be an unob- 
trusive experience. 

Granting, then, that the magnitude and direction Magnitude 

. ...... and direc- 

of forces impressed on a point alone are decisive, it will on the sole 

decisive 

be perceived that two equal and opposite forces, as they factors, 
cannot uniquely and precisely determine any motion, 
are in equilibrium. So, also, at 
right angles to its direction, a 
force/ is unable uniquely to de- 
termine a motional effect. But 

if a force/ is inclined at an an- 
gle to another direction ss (Fig. 

36), it is able to determine a mo- 

tion in that direction. Yet ex- " 

perience alone can inform us, 

that the motion is determined in the direction of s's 

and not in that of ss' ; that is to say, in the direction 

of the side of the acute angle or in the direction of the 

projection of / on s's. 

Now this latter experience is made use of by Ber-The^rfoi 

, , , direction 

noulh at the very start. The sense, namely, ot the re- derivable 

.... only from 

sultant of two equal forces acting at right angles to one experience, 
another is obtainable only on the ground of this expe- 
rience. From the principle of symmetry follows only, 
that the resultant falls in the plane of the forces and 
coincides with the line of bisection of the angle, not 
however that it falls in the acute angle. But if we sur- 
render this latter determination, our whole proof is ex- 
ploded before it is begun. 

10. If, now, we have reached the conviction that 
our knowledge of the effect of the direction of a force is 



44 THE SCIENCE OE MECHANICS. 

So also solely obtainable from experience, still less then shall 

must the ..... 

form of the we believe it in our power to ascertain by any other way 
thusde- the form of this effect. It is utterly out of our power, 
to divine, that a force p acts in a direction j that makes 
with its own direction the angle a, exactly as a force 
/ cos a in the direction s ; a statement equivalent to the 
proposition of the parallelogram of forces. Nor was 
it in Bernoulli's power to do this. Nevertheless, he 
makes use, scarcely perceptible it is true, of expe- 
riences that involve by implication this very mathe- 
matical fact. 
The man- A person already familiar with the composition 

which the and resolution of forces is well aware that several forces 

assump- 
tions men- acting at a point are, as regards their effect, replaceable, 

tioned enter. r ... 

into Ber- in every respect and in every direction, by a single force. 

noulli's de- . , . J 

duction. This knowledge, in Bernoulli's mode of proof, is ex- 
pressed in the fact that the forces p, q are regarded as 
absolutely qualified to replace in all respects the forces 
s, u and /, v, as well in the direction of r as in every 
other direction. Similarly r is regarded as the equiv- 
alent of / and q. It is further assumed as wholly in- 
different, whether we estimate s, u, t, v first in the 
directions of/, q, and then/, q in the direction of r, or 
s, u, t, v be estimated directly and from the outset in 
the direction of r. But this is something that a person 
only can know who has antecedently acquired a very 
extensive experience concerning the composition and 
resolution of forces. We reach most simply the knowl- 
edge of the fact referred to, by starting from the knowl- 
edge of another fact, namely that a force / acts in a 
direction making with its own an angle a, with an effect 
equivalent to p cos a. As a fact, this is the way the 
perception of the truth was reached. 

Let the coplanar forces P, P', P". . . be applied to 



THE PRINCIPLES OF STATICS. 45 

one and the same point at the angles a, a, a" . . . with Mathemat- 
ical analy- 
a given direction X. These forces, let us suppose, are sis of the 

... . results of 

replaceable by a single force 77, which makes with A'thetmeand 

" .... . necessary 

an angle //. By the familiar principle we have then assumption. 

2Pcosa = 77 cos //. 
If 77 is still to remain the substitute of this system of 
forces, whatever direction X may take on the system 
being turned through any angle 6, we shall further 
have 

2P cos (a + S) = 77 cos (// -f S), 
or 

cosar 77cos//) cost? (2Psina 77 sin//) sinrf = 0. 
If we put 

2P cosar 77 cos;/ = A, 
(2Psina 77 sin//) = B, 
B 

it follows that 

A costf -f B sintf = V A* + i?* sin ((? + r) = 0, 
which equation can subsist for every S only on the con- 
dition that 

A = 2Pcosa 77 cos// = 

and 

B = (2Psina 77 sin//) = ; 

whence results 

77 cos// = 2P cosar 

77 sin// = ^Ps'ma. 
From these equations follow for 77 and // the deter- 
minate values 



and 

2Psina 

tan// = =-= -. 

2.P cosar 



46 THE SCIENCE OF MECHANICS. 

The actual Granting, therefore, that the effect of a force in every 

results not ... , , , . . . ,. 

deducibie direction can be measured by its projection on that di- 

on any . . , 

other sup- rection, then truly every system of forces acting at a 

position. . . , , , , r i 

point is replaceable by a single force, determinate in 
magnitude and direction. This reasoning does not hold, 
however, if we put in the place of cos a any general func- 
tion of an angle, <p (a). Yet if this be done, and we still 
regard the resultant as determinate, we shall obtain for 
cp(a), as may be seen, for example, from Poisson's 
deduction, the form cos a. The experience that several 
forces acting at a point are always-, in every respect, 
replaceable by a single force, is therefore mathemat- 
ically equivalent to the principle of the parallelogram 
of forces or to the principle of projection. The prin- 
ciple of the parallelogram or of projection is, how- 
ever, much easier reached by observation than the 
General re- more general experience above mentioned by statical 
observations. And as a fact, the principle of the par- 
allelogram was reached earlier. It would require in- 
deed an almost superhuman power of perception to 
deduce mathematically, without the guidance of any 
further knowledge of the actual conditions of the ques- 
tion, the principle of the parallelogram from the gen- 
eral principle of the equivalence of several forces to a 
single one. We criticise accordingly in the deduction 
of Bernoulli this, that that which is easier to observe 
is reduced to that which is more difficult to observe. 
This is a violation of the economy of science. Bernoulli 
is also deceived in imagining that he does not proceed 
from any fact whatever of observation. 
An addi- We must further remark that the fact that the forces 

sumption of are independent of one another, which is involved in 

Bernoulli. , , - . ; ... . ., 

the law of their composition, is another experience 
which Bernoulli throughout tacitly employs. As long 



THE PRIXCIPLES OF STATICS. 47 

as we have to do with uniform or symmetrical systems 
of forces, all equal in magnitude, each can be affected 
by the others, even if they are not independent, only 
to the same extent and in the same way. Given but 
three forces, however, of which two are symmetrical 
to the third, and even then the reasoning, provided 
we admit that the forces may not be independent, pre- 
sents considerable difficulties. 

11. Once we have been led, directly or indirectly. Discussion 
to the principle of the parallelogram of forces, once we acter of the 
have perceived it, the principle is just as much an ob- 
servation as any other. If the observation is recent, it 

of course is not accepted with the same confidence as 
old and frequently verified observations. We then seek 
to support the new observation by the old, to demon- 
strate their agreement. By and by the new observa- 
tion acquires equal standing with the old. It is then 
no longer necessary constantly to reduce it to the lat- 
ter. Deduction of this character is expedient only in 
cases in which observations that are difficult directly 
to obtain can be reduced to simpler ones more easily 
obtained, as is done with the principle of the parallel- 
ogram of forces in dynamics. 

12. The proposition of the parallelogram of forces Experimen- 
has also been illustrated by experiments especially tion of the 
instituted for the purpose. An apparatus very wellacontriv- 
adapted to this end was contrived by Cauchy. The Cauchy. 
centre of a horizontal divided circle (Fig. 37) is marked 

by a pin. Three threads/,/*,/", tied together at a 
point, are passed over grooved wheels r, r ', r", which 
can be fixed at any point in the circumference of the 
circle, and are loaded by the weights /, /', /". If three 
equal weights be attached, for instance, and the wheels 
placed at the marks of division o, 120, 240, the point at 



48 



THE SCIENCE OF MECHANICS. 



Experimen- which the strings are knotted will assume a position 

tal illustra- . . .... _. , 

tionofthe just above the centre of the circle. Ihree equal forces 

principle. . , ,. , . 

acting at angles of 120 , accordingly, are in equilib- 



rium. 




Fig. 37. 

If we wish to represent another and different case, 
we may proceed as follows. We imagine any two 
forces p, q acting at any angle a, represent (Fig. 38) 
them by lines, and construct on them as sides a paral- 
lelogram. We supply, further, a force 
equal and opposite to the resultant r. 
The three forces p, q, r hold each 
other in equilibrium, at the angles vis- 
ible from the construction. We now 
place the wheels of the divided circle on 
the points of division o, a, a -f- /3, and 
load the appropriate strings with the 
weights/, q, r. The point at which the 
strings are knotted will come to a position exactly 
above the middle point of the circle. 




Fig. 38. 



THE PRIXCIPLES OF STATICS. 49 



THE PRINCIPLE OF VIRTUAL VELOCITIES. 

i. We now pass to the discussion of the principle The truth 

r of the prin- 

of virtual (possible) displacements.* The truth of cipie first 

vt ' * remarked 

this principle was first remarked by Stevinus at the by stevinos 
close of the sixteenth century in his investigations on 
the equilibrium of pulleys and combinations of pulleys. 
Stevinus treats combinations of pulleys in the same 
way they are treated at the present day. In the case 

Termed in English the principle of " virtual velocities," this being the 
original phrase yvitetse virttutlt) introduced by John Bernoulli. See the 
text, page 56. The word virtuaiis seems to have been the fabrication of Duns 
Scotus (see the Century Dictionary, under virtual); but virtualiter was used 
by Aquinas, and virtus had been employed for centuries to translate dvvafug, 
and therefore as a synonym for fotentia. Along with many other scholastic 
terms, virtual passed into the ordinary vocabulary of the English language. 
Everybody remembers the passage in the third book of Paradise Lost, 
" Love not the beaT'nly Spirits, and bow thir Love 

Express they, by looks oaely. or do they mix 

Irradiaace, virtual or immediate touch ? " Milton. 

So, we all remember how it was claimed before our revolution that America 
had " virtual representation " in parliament. In these passages, as in Latin, 
virtual means : existing in effect, but not actually. In the same sense, the 
word passed into French ; and was made pretty common among philosophers 
by Leibnitz. Thus, he calls innate ideas in the mind of a child, not yet brought 
to consciousness, "des connoissances virtuelles." 

The principle in question was an extension to the case of more than two 
forces of the old rule that "what a machine gains in funster, it loses in velocity.'* 
Bernoulli's modification reads that the sum of the products of the forces into 
their virtual velocities must vanish to give equilibrium. He says, in effect : 
give the system any possible and infinitesimal motion you please, and then 
the simultaneous displacements of the points of application of the forces, 
resolved in the directions of those forces, though they are not exactly velocities, 
since they are only displacements in one time, are, nevertheless, virtually 
velocities, for the purpose of applying the rule that what a machine gains in 
power, it loses in velocity. 

Thomson and Tait say : " If the point of application of a force be dis- 
placed through a small space, the resolved part of the displacement in the di- 
rection of the force has been called its I'irtual Velocity. This is positive or 
negative according as the virtual velocity is in the same, or in the opposite, 
direction to that of the force." This agrees with Bernoulli's definition which 
may be found in Varignon's Xouvelle micanique. Vol. II, Chap. ix. Trans. 



o 



THE SCIENCE OF MECHANICS. 



Stevinus-s a (Fig. 39) equilibrium obtains, when an equal weight P 
tionsontheis suspended at each side, for reasons already familiar. 

equilibrium 

of pulleys. In b, the weight P is suspended by two parallel cords, 



o 



n n 
p p 




A 












8 






each of which accordingly supports the weight P/i, 
with which weight in the case of equilibrium the free 
end of the cord must also be loaded. In c, P is sus- 
pended by six cords, and the weighting of the free ex- 
tremity with P/6 will accordingly produce equilibrium. 
In d, the so-called Archimedean or potential pulley,* P 
in the first instance is suspended by two cords, each 
of which supports P/t. ; one of these two cords in turn 
is suspended by two others, and so on to the end, so 
that the free extremity will be held in equilibrium by 
the weight P/8. If we impart to these assemblages 
of pulleys displacements corresponding to a descent of 
the weight P through the distance h, we shall observe 
that as a result of the arrangement of the cords 

a distance h in a 
" 2/1 " b 
" 6/1 " c \ 
8/1 " d 



the counterweight P 
" Pji 

it pj% 



will ascend 



* These terms are not in use in English. Trans. 



THE PRINCIPLES OF STATICS. 



5i 



In a system of pulleys in equilibrium, therefore, His conciu- 

. . sions the 

the products of the weights into the displacements germ of the 

1 -i 1 ,- T - principle. 

they sustain are respectively equal. (" Ut spatium 
agentis ad spatium patientis, sic potentia patientis ad 
potentiam agentis." Stevini, Hypomnemata, T. IV, 
lib. 3, p. 172.) In this remark is contained the germ 



of the principle of virtual displacements. - . . 
2. GALiLEO.recognjsed the truth of 



On case of the 
inclined 
plane. 




Fig. 40. 



n orthe principle in Galileo's 

n recognition 

another case, and that a somewhat more general one : of the prin- 

1 i- .i i- j i ^ cipleinthe 

namely, in its application to the inclined plane, 
an inclined plane (Fig. 40), 
the length of which AB is 
double the height BC, a load 
Q placed on AB is held in 
equilibrium by the load P act- 
ing along the height BC, if 
P Q 2. If the machine be 
set in motion, P = Q/2 will descend, say, the vertical 
distance //, and Q will ascend the same distance // along 
the incline AB. Galileo, now, allowing the phenom- 
enon to exercise its full effect on his mind, perceives, 
that equilibrium is determined not by the weights 
alone but also by their possible approach to and reces- 
sion from the centre of the earth. Thus, while Q'/2 de- 
scends along the vertical height the distance //, Q as- 
cends // along the inclined length, vertically, however, 
only h/2 ; the result being that the products Q(h/2) 
and (Q/2)h come out equal on both sides. The eluci- 
dation that Galileo's observation affords and the light character 

j-rr -ii 1 f Galileo's 

it diffuses, can hardly be emphasised strongly enough, observation 
The observation is so natural and unforced, moreover, 
that we admit it at once. What can appear simpler 
than that no motion takes place in a system of heavy 



52 THE SCIENCE OF MECHANICS. 

bodies when on the whole no heavy mass can descend. 
Such a fact appears instinctively acceptable. AdA . 
comparison Galileo's conception of the inclined plane strikes 

of it with . * 

thatofste- us as much less ingenious than that of Stevinus, but 
v ' nus - 

we recognise it as more natural and more profound. It 

is in this fact that Galileo discloses such scientific great- 
ness : that he had the intellectual audacity to see, in a 
subject long before investigated, more than his prede- 
cessors had seen, and to trust to his own perceptions. 
With the frankness that was .characteristic of him he 
unreservedly places before the reader his own view, 
together with the considerations that led him to it. 
TheTorri- 3. Torricelli, by the employment of the notion of 

cellian J . J . 

form of the "centre of gravity," has put Galileo's principle in a 
form in which it appeals still more to our instincts, but 
in which it is also incidentally applied by Galileo him- 
self. According to Torricelli equilibrium exists in a 
machine when, on a displacement being imparted to it, 
the centre of gravity of the weights attached thereto 
cannot descend. On the supposition of a displacement 
in the inclined plane last dealt with, P, let us say, de- 
scends the distance //, in compensation wherefor Q 
vertically ascends h . sin a. Assuming that the centre 
of gravity does not descend, we shall have 

P.hQ.h sin a . 

h-t 7^ Q, or P. /i O. h sin a = 0, 

P -\- Q 

or 

P=-Qs\na=Q^. 

If the weights bear to one another some different pro- 
portion, then the centre of gravity can descend when a 
displacement is made, and equilibrium will not obtain. 
We expect the state of equilibrium instinctively, when j 
the centre of gravity of a system of heavy bodies can- j 



THE PRIXCIPLES OF STATICS. 53 

not descend. The Torricellian form of expression, how- 
ever, contains in no respect more than the Galilean. 

x. As with systems of pulleys and with the inclined The appii- 

~ J * . cation of 

Diane, so also the validity of the principle of virtual the princi- 
displacements is easily demonstrable for the other ma- other ma- 

r * chines. 

chines : for the lever, the wheel and axle, and the rest. 
In a wheel and axle, for instance, with the radii R, r 
and the respective weights P, Q, equilibrium exists, 
as we know, when PR = Qr. If we turn the wheel 
and axle through the angle a, P will descend Ra, and 
Q will ascend ra. According to the conception of 
Stevinus and Galileo, when equilibrium exists, P. Ra 
z=z Q. ra, which equation expresses the same thing as 
the preceding one. 

5. When we compare a system of heavy bodies in The cme- 
which motion is taking place, with a similar system state of 

r equilibrium 

which is in equilibrium, the question forces itself upon 
us : What constitutes the difference of the two cases? 
What is the factor operative here that determines mo- 
tion, the factor that disturbs equilibrium, the factor 
that is present in the one case and absent in the other? 
Having put this question to himself, Galileo discovers 
that not only the weights, but also the distances of 
their vertical descents (the amounts of their vertical 
displacements) are the factors that determine motion. 
Let us call P, P', P" . . . the weights of a system of 
heavy bodies, and h, h', h" . . . their respective, simul- 
taneously possible vertical displacements, where dis- 
placements downwards are reckoned as positive, and 
displacements upwards as negative. Galileo finds 
then, that the criterion or test of the state of equilib- 
rium is contained in the fulfilment of the condition 
Ph + Fh' -f P" h" + . . . = 0. The sum Ph -f- P'h' 
-j- P"h"-\- ... is the factor that destroys equilibrium, 



54 THE SCIENCE OF MECHANICS. 

the factor that determines motion. Owing to its im- 
portance this sum has in recent times been character- 
ised by the special designation work. 
There is no 6. Whereas the earlier investigators, in the compari- 

necessity in 

our choice son of cases of equilibrium and cases of motion, directec 

of the cri- . . . . 

teria. their attention to the weights and their distances troi 

the axis of rotation and recognised the statical mo- 
ments as the decisive factors involved, Galileo fixes 
his attention on the weights and their distances of de 
scent and discerns work as the decisive factor involved. 
It cannot of course be prescribed to the inquirer 
what mark or criterion of the condition of equilibriui 
he shall take account of, when several are present t( 
choose from. The result alone can determine whether 
his choice is the right one. But if we cannot, for rea- 
Andaiiare sons already stated, regard the significance of the stat- 
from the ical moments as given independently of experience, as 
source. something self-evident, no more can we entertain this 
view with respect to the import of work. Pascal errs, 
and many modern inquirers share this error with him, 
when he says, on the occasion of applying the principle 
of virtual displacements to fluids: "Etant clair que e'est 
la merae chose de faire faire un pouce de chemin a cent 
livres d'eau, que de faire faire cent pouces de chemin 
a une livre d'eau." This is correct only on the suppo- 
sition that we have already come to recognise work as 
the decisive factor ; and that it is so is a fact which 
experience alone can disclose, 
illustration If we have an equal-armed, equally-weighted lever 
ceding re- before us, we recognise the equilibrium of the lever as 
the only effect that is uniquely determined, whether we 
regard the weights and the distances or the weights 
and the vertical displacements as the conditions that 
determine motion. Experimental knowledge of this 






THE PRINCIPLES OF STATICS. 



55 



or a similar character must, however, in the necessity of 
the case precede any judgment of ours with regard to 
the phenomenon in question. The particular way in 
which the disturbance of equilibrium depends on the 
conditions mentioned, that is to say, the significance 
of the statical moment {PL) or of the work (Ph), is 
even less capable of being philosophically excogitated 
than the general fact of the dependence. 

7. When two equal weights with equal and op- Reduction 
posite possible displacements are opposed to each erai case of 

i_ , i_ r -i-i theprinci- 

other, we recognise at once the subsistence of equilib-pieto the 

ttt , , , , 1 simpler and 

num. \\ e might now be tempted to reduce the more special case 
general case of the weights P, P' with the capacities of 
displacement//,//', where 
P/i = P'h\ to the sim- 
pler case. Suppose we 
have, for example, (Fig. 
41) the weights 3 P and 
4 P on a wheel and axle 
with the radii 4 and 3. 
We divide the weights 
into equal portions of the 

definite magnitude P, which we designate by a, b, c, 
d> e > J> g- We then transport a, b, c to the level -f- 3, 
and d, e, f to the level 3. The weights will, of 
themselves, neither enter on this displacement nor 
will they resist it. We then take simultaneously the 
weight g at the level and the weight a at the level 
-f- 3, push the first upwards to 1 and the second 
downwards to + 4> then again, and in the same way, 
g to 2 and b to + 4, g to 3 and c to -f 4. To all 
these displacements the weights offer no resistance, 
nor do they produce them of themselves. Ultimately, 
however, a, b, c (or iP) appear at the level + 4 and 











-J 

-2 

--/ 


a b c 
















+ 2 i 







a 


7 < /V 


+ 3 

+ 4 . 









F.g 41. 



56 THE SCIENCE OF MECHANICS. 

The gen- d, e, f, g (or 4.P) at the level 3. Consequently, 
with respect also to the last-mentioned total displace- 
ment, the weights neither produce it of themselves 
nor do they resist it ; that is to say, given the ratio of 
displacement here specified, and the weights will be 
in equilibrium. The equation 4.3/* 3. 4^=0 is, 
therefore, characteristic of equilibrium in the case as- 
sumed. The generalisation (PA P'h' = 0) is ob- 
vious. 
Thecondi- If we carefully examine the reasoning of this case, 
character we shall quite readily perceive that the inference in- 
ence. ' volved cannot be drawn unless we take for granted 
that the order of the operations performed and the path 
by which the transferences are effected, are indifferent, 
that is unless we have previously discerned that work 
is determinative. We should commit, if we accepted 
this inference, the same error that Archimedes com- 
mitted in his deduction of the law of the lever ; as has 
been set forth at length in a preceding section and j 
need not in the present case be so exhaustively dis- 
cussed. Nevertheless, the reasoning we have pre-; 
sented is useful, in the respect that it brings palpably! 
home to the mind the relationship of the simple and| 
the complicated cases. 
Theuniver- 8. The universal applicability of the principle of 
b?iif y P of 'the virtual displacements to all cases of equilibrium, \va:| 
firstper- e perceived by John Bernoulli ; who communicated hi j 
John Be* discovery to Varignon in a letter written in 171 7. W 



noulli. 



will now enunciate the principle in its most genera 
form. At the points A, B, C . . . (Fig. 42) the force 
P, P', P" . . . are applied. Impart to the points an 
infinitely small displacements v, v' , v" . . . compatib.' 
with the character of the connections of the points (s< 
called virtual displacements), and construct the pn 




THE PRIXCIPLES OF STATICS. 57 

iections p, p\ p" of these displacements on the direc- General 

1 ' * * . enunciation 

tions of the forces. These projections we consider of the prin- 
ciple, 
positive when they fall in . 

the direction of the force, J^ r^ - 

and negative when they fall 

in the opposite direction. 

The products Pp, P'p', 

P"p", . . . are called virtual 

moments, and in the two 

cases just mentioned have F>g-4*. 

contrary signs. Now, the principle asserts, that for the 

case of equilibrium Pp -f P'p' -\- P" p" + . . . = 0, or 

more briefly 2Pp = 0. 

q.- Let us now examine a few points more in detail. Detailed 

-11 examina- 

Previous to Newton a force was almost universally "on of the 

principle. 

conceived simply as the pull or the pressure of a heavy 
body. The mechanical researches of this period dealt 
almost exclusively with heavy bodies. When, now, 
in the Newtonian epoch, the generalisation of the idea 
of force was effected, all mechanical principles known 
to be applicable to heavy bodies could be transferred 
at once to any forces whatsoever. It was possible to 
replace every force by the pull of a heavy body on a 
string. In this sense we may also apply the principle 
of virtual displacements, at first discovered only for 
heavy bodies, to any forces whatsoever. 

Virtual displacements are displacements consistent Definition 
with the character of the connections of a system and dispiace- 

. ments. 

with one another. If, for example, the two points of 
a system, A and B, at which forces act, are connected 
(Fig. 43, 1) by a rectangularly bent lever, free to re- 
volve about C, then, if CB = 2 CA, all virtual dis- 
placements of B and A are elements of the arcs of cir- 
cles having C as centre ; the displacements of B axe 



1 



58 THE SCIENCE OF MECHANICS. 

always double the displacements of A, and both are in 
every case at right angles to each other. If the points 
A, B (Fig. 43, 2) be connected by a thread of the length 

r /, adjusted to slip through 

A \ stationary rings at C and D, 

I \ then all those displacements 

q 2 f A and B are virtual in 

Fig. 43. which the points referred to 

move upon or within two spherical surfaces described 

with the radii r 1 and r 2 about C and D as centres, 

where r l -f- ^ 2 ~f~ CD = ^ 

The reason The use of infinitely small displacements instead of 

for the use _.... . .., .... 

of \i\hxy\\.z\y finite displacements, such as Galileo assumed, is justi- 
piacements. fied by the following consideration. If two weights 
are in equilibrium on an inclined plane (Fig. 44), the 
equilibrium will not be disturbed if the inclined plane, 
at points at which it is not in immediate contact with 
the bodies considered, passes into 
a surface of a different form. The 
essential condition is, therefore, 
the momentary possibility of dis- 
Fig. 44. placement in the momentary con- 

figuration of the system. To judge of equilibrium we 
must assume displacements vanishingly small and such 
only ; as otherwise the system might be carried over 
into an entirely different adjacent configuration, for 
which perhaps equilibrium would not exist. 
a limita- That the displacements themselves are not decisive 

but only the extent to which they occur in the direc- 
tions of the forces, that is only their projections on the 
lines of the forces, was, in the case of the inclined plane, 
perceived clearly enough by Galileo himself. 

With respect to the expression of the principle, it 
will be observed, that no problem whatever is presented 




tion 



THE PRIXCIPLES OF STATICS. 



59 



if all the material points of the system on which forces General re- 
act. are independent of each other. Each point thus 
conditioned can be in equilibrium only in the event 
that it is not movable in the direction in which the force 
acts. The virtual moment of each such point vanishes 
separately. If some of the points be independent of 
each other, while others in their displacements are de- 
pendent on each other, the remark just made holds 
good for the former ; and for the latter the fundamental 
proposition discovered by Galileo holds, that the sum 
of their virtual moments is equal to zero. Hence, the 
sum-total of the virtual moments of all jointly is equal 
to zero. 

10. Let us now endeavor to get some idea of the Examples, 
significance of the principle, by the consideration of a 
few simple examples that cannot be 
dealt with by the ordinary method 
of the lever, the inclined plane, and 
the like. 

The differential pulley of Wes- 
ton (Fig. 45) consists of two coax- 
ial rigidly connected cylinders of 
slightly different radii r t and r 2 
<C*V A cord or chain is passed 
round the cylinders in the manner 
indicated in the figure. If we pull 
in the direction of the arrow with 
the force P, and rotation takes place 
through the angle <p, the weight Q attached below will 
be raised. In the case of equilibrium there will exist 
between the two virtual moments involved the equa- 
tion 




Fig. 45- 



e^ 



.. 



<p = Pr x <p, or P = Q 



2r, 



6o 



THE SCIENCE OF MECHANICS. 



A suspend- 
ed wheel 
and axle. 







A double 
cylinder on 
a horizon- 
tal surface 



Roberval's 
balance. 



A wheel and axle of weight Q (Fig. 46), which on 
the unrolling of a cord to which the weight P is at- 
tached rolls itself up on a second cord 
wound round the axle and rises, gives 
for the virtual moments in the case of 
equilibrium the equation 

P(P r) (p= Qrcp, or P = r . 

P t 

In the particular case R r = 0, we 
must also put, for equilibrium, Qr = 0, or, 
for finite values of r, (? = 0. In reality the 
string behaves in this case like a loop in 
which the weight Q is placed. The lat- 
ter can, if it be different from zero, continue to roll itself 
downwards on the string without moving the weight P. 
If, however, when P r, we also put Q = 0, the re- 
sult will be P=Q-, an indeterminate value. As a mat- 
ter of fact, every weight P holds the apparatus in equi- 
librium, since when P = r none can possibly descend. 
A double cylinder (Fig. 47) of the radii r, R lies with 
friction on a horizontal surface, and a force Q is brought 

to bear on the string at- 
tached to it. Calling the re- 
sistance due to friction P, 
equilibrium exists when 

"mmzmmztm p=(p7/p) Q . u p > 

Fig- 47. (P^r/P) Q, the cylinder, 

on the application of the force, will roll itself up on 
the string. 

Roberval's Balance (Fig. 48) consists of a paral- 
lelogram with variable angles, two opposite sides of , 
which, the upper and lower, are capable of rotation j 
about their middle points A, B. To the two remaining 
sides, which are always vertical, horizontal rods are 




THE PRINCIPLES OF STATICS. 



61 



fastened. If from these rods we suspend two equal 

weights P, equilibrium will subsist independently of 

the position of the points 

of suspension, because on 

displacement the descent 

of the one weight is always 

equal to the ascent of the 

other. 

At three fixed points A, 
B, C (Fig. 49) let pulleys 
be placed, over which three strings are passed loaded Jhree knot- 
with equal weights and knotted at O. In what posi- 1 " 
tion of the strings will equilibrium exist? We will call 
the lengths of the three strings AO = s x , BO = s 2 , 




Fig. 48, 





Fig. 49- F>g- 5- 

CO = s 3 . To obtain the equation of equilibrium, let 
us displace the point O in the directions s 2 and s 3 the 
infinitely small distances 6s 2 and 6s 3 , and note that by 
so doing even* direction of displacement in the plane 
ABC (Fig. 50) can be produced. The sum of the vir- 
tual moments is 



P8s 2 P8s 2 cos a -f- P6s 2 cos (a -f- /?) 
+ ^^3 P6s 3 cos/? -f- PSs 3 cos (a + /S) 



u 



or 



[1 cos a -\- cos (a + /?)] S s 2 -f- [1 cos /? 
+ cos (a + /?)] oV 3 = 0. 
But since each of the displacements 6V 2 , ds 3 is ar- 



62 THE SCIENCE OF MECHANICS. 

bitrary, and each independent of the other, and may by 
themselves be taken = 0, it follows that 

1 cos a -}- cos (a -\- /3) = 
1 cos /? + cos O + /?) = 0. 

Therefore 

cos a = cos /?, 

and each of the two equations may be replaced by 

1 cos a -\- cos 2a = 0; 



or cos' = 



2- 



wherefore + = 120. 

Remarks on Accordingly, in the case of equilibrium, each of the . 

ingcase. strings makes with the others angles of 120 ; which is, 
moreover, directly obvious, since three equal forces can 
be in equilibrium only when such an arrangement ex- 
ists. This once known, we may find the position of 
the point O with respect to ABC in a number of dif- 
ferent ways. We may proceed for instance as follows. 
We construct on AB, BC, CA, severally, as sides, 
equilateral triangles. If we describe circles about these 
triangles, their common point of intersection will be j 
the point O sought ; a result which easily follows from | 
the well-known relation of the angles at the centre and 
circumference of circles. 

The case of A bar OA (Fig. 51) is revolvable about O in the j 

a bar re- . . _ , ..,. 

voivabie plane of the paper and makes with a fixed straight line ! 

about one _ , . , 

of us ex- rf OX the variable angle ! 

~~p a. At A there is ap- 
plied a force P which j 
^? makes with OX the 

X angle y, and at B, on 
Fig. 51. a ring displaceable 

along the length of the bar, a force Q, making with 
OX the angle fi. We impart to the bar an infinitely 



tremities. 




THE PRINCIPLES OF STATICS. 63 

small rotation, in consequence of which B and A move The case of 

, a bar re- 

forward the distances 6s and 6s at right angles to OA, voivabie 

i- 1 it t> 1 about one 

and we also displace the ring the distance or along the of its ex- 

_, ..... _ -ii 11 1 tremities. 

bar. The variable distance OB we will call r, and we 
will let OA = a. For the case of equilibrium we have 
then 

Q6r cos (J3a)+ Q6s sin (J3 a) + 
P6s 1 sin (a y) = 0. 

As the displacement 6r has no effect whatever on 
the other displacements, the virtual moment therein 
involved must, by itself, = 0, and since 6r may be of 
any magnitude we please, the coefficient of this virtual 
moment must also = 0. We have, therefore, 

Q cos (J3 a) = 0, 
or when Q is different from zero, 
pa = 90. 

Further, in view of the fact that 6s x =(a/r) 6s, we 
also have 

rQ sin (J3 a) -f a P sin (a y) = 0, 
or since sin (ft a) = 1 , 

rQ + a P sin (a y) = ; 
wherewith the relation of the two forces is obtained. 

11. An advantage, not to be overlooked, which Every gen- 
every general principle, and therefore also the prin- cipie in- 
ciple of virtual displacements, fur- 



nishes, consists in the fact that it 
saves us to a great extent the ne- 1 
cessity of considering every new par- * 1 
ticular case presented. In the posses- 
sion of this principle we need not, for Fig. 52. 
example, trouble ourselves about the details of a ma- 
chine. If a new machine say were so enclosed in a 



I economy of 
k' thought. 



6 4 



THE SCIENCE OF MECHANICS. 






Further re- 
marks on 
the general 
expression 
of the prin- 
ciple. 




Fig- 53- 



Modifica- 
tion of the 
previous 
equation of 
condition. 



box (Fig. 52), that only two levers projected as points 
of application for the force P and the weight P', and I 
we should find the simultaneous displacements of these 
levers to be h and h', we should know immediately that 
in the case of equilibrium Ph = P' //', whatever the 
construction of the machine might be. Every principle I 
of this character possesses therefore a distinct econom- 
ical value. 

12. We return to the general expression of the prin- 
ciple of virtual displacements, in order to add a few 

further remarks. Ifi 
at the points A, B, 
C . . . . the forces! 
P,P',P". . . .act,! 
and /, p', p" . . . \ 
are the projections j 
of infinitely smal 
mutually compatible displacements, we shall have fo: 
the case of equilibrium 

Pp + P'P' + P'P" + = 0. 
If we replace the forces by strings which pass ove 
pulleys in the directions of the forces and attach theret 
the appropriate weights, this expression simply a: 
serts that the centre of gravity of the system of weigh 
as a whole cannot descend. If, however, in certain di 
placements it were possible for the centre of gravi 
to rise, the system would still be in equilibrium, as tJ| 
heavy bodies would not, of themselves, enter on a ' 
such motion. In this case the sum above given woi I 
be negative, or less than zero. The general expressi 1 
of the condition of equilibrium is, therefore, 
Pp + Pp' + P"p" + . . . < 0. 
When for every virtual displacement there exi s 



THE PRIXCIPLES OF STATICS. 65 

another equal and opposite to it, as is the case for ex- 
ample in the simple machines, we may restrict ourselves 
to the upper sign, to the equation. For if it were pos- 
sible for the centre of gravity to ascend in certain 
displacements, it would also have to be possible, in 
consequence of the assumed reversibility of all the vir- 
tual displacements, for it to descend. Consequently, 
in the present case, a possible rise of the centre of 
gravity is incompatible with equilibrium. 

The question assumes a different aspect, however, The condi- 
when the displacements are not all reversible. Two the sum of 
bodies connected together by strings can approach moments 
each other but cannot recede from each other beyond equal to or 
the length of the strings. A body is able to slide or zero, 
roll on the surface of another body ; it can move away 
from the surface of the second body, but it cannot 
penetrate it. In these cases, therefore, there are dis- 
placements that cannot be reversed. Consequently, 
for certain displacements a rise of the centre of gravity 
may taka place, while the contrary displacements, to 
which the descent of the centre of gravity corresponds, 
are impossible. We must therefore hold fast to the 
more general condition of equilibrium, and say, the sum 
of the virtual moments is equal to or less than zero. 

13. Lagrange in his Analytical Mechanics attempted The La- 
a deduction of the principle of virtual displacements, deduction 

. . . . of the prin- 

which we will now consider. At the points A, B, cipie. 
C . . . . (Fig. 54) the forces P, P', P" . . . . act. We 
imagine rings placed at the points in question, and 
other rings A', B" , C . . . . fastened to points lying in 
the directions of the forces. We seek some common 
measure Q/2 of the forces P, P' , P" .... that enables 
us to put : 



66 



THE SCIENCE OF MECHANICS. 



Effected by 
means of a 
set of pul- 
leys and a 
single 
weight 



2n 



2ri 



2". 



= P", 



where n, ri, n" . . . . are whole numbers. Further, we 
make fast to the ring A' a string, carry this string back 
and forth n times between A' and A, then through B ', 




n' times back and forth between B and B, then through 
C, n" times back and forth between C and C, and, 
finally, let it drop at C, attaching to it there the weight 
Q/2. As the string has, now, in all its parts the ten- 
sion Q/2, we replace by these ideal pulleys all the 
forces present in the system by the single force Q/2. 
If then the virtual (possible) displacements in any given 
configuration of the system are such that, these dis- 
placements occurring, a descent of the weight Q/2 can 
take place, the weight will actually descend and pro- 
duce those displacements, and equilibrium therefore 
will not obtain. But on the other hand, no motion 
will ensue, if the displacements leave the weight Q/2 
in its original position, or raise it. The expression of 
this condition, reckoning the projections of the virtual 
displacements in the directions of the forces positive, 



THE PRLXCIPLES OF STATICS. 67 

and having regard for the number of the turns of the 
string in each single pulley, is 

2np + '2n'p' + Iri'p" + . . . < 0. 

Equivalent to this condition, however, is the ex- 
pression 



or 



In ff - 'In I / + 2n" fjT + . . . < 0, 



Pp + rp' + P"P" + < 0. 



14. The deduction of Lagrange, if stripped of the The con- 
rather odd fiction of the pulleys, really possesses con- turesof La- 

1 grange's 

vmcing features, due to the fact that the action of a deduction, 
single weight is much more immediate to our expe- 
rience and is more easily followed than the action of 
several weights. Yet it is not proved by the Lagrangian 
deduction that work is the factor determinative of the 
disturbance of equilibrium, but is, by the employment 
of the pulleys, rather assumed by it. As a matter of 
fact every pulley involves the fact enunciated and rec- 
ognised by the principle of virtual displacements. The 
replacement of all the forces by a single weight that 
does the same work, presupposes a knowledge of the 
import of work, and can be proceeded with on this as- 
sumption alone. The fact that some certain cases are it is not, 
more familiar to us and more immediate to our expe- proof, 
rience has as a necessary result that we accept them 
without analysis and make them the foundation of our 
deductions without clearly instructing ourselves as to 
their real character. 

It often happens in the course of the development 
of science that a new principle perceived by some in- 
quirer in connection with a fact, is not immediately 
recognised and rendered familiar in its entire generality. 



68 THE SCIENCE OF MECHANICS. 

Theexpe- Then, every expedient calculated to promote these 

dients em- . . * . 

ployed to ends, is, as is proper and natural, called into service, 
new prin- All manner of facts, in which the principle, although 
contained in them, Jias not yet been recognised by in- 
quirers, but which from other points of view are more 
familiar, are called in to furnish a support for the new 
conception. It does not, however, beseem mature 
science to allow itself to be deceived by procedures of 
this sort. If, throughout all facts, we clearly see and dis- 
cern a principle which, though not admitting of proof, 
can yet be known to prevail, we have advanced much 
farther in the consistent conception of nature than if 
we suffered ourselves to be overawed by a specious 
Value of the demonstration. If we have reached this point of view, 
proof. we shall, it is true, regard the Lagrangian deduction 
with quite different eyes ; yet it will engage neverthe- 
less our attention and interest, and excite our satis- 
faction from the fact that it makes palpable the simi- 
larity of the simple and complicated cases. 

15. Maupertuis discovered an interesting proposi- 
tion relating to equilibrium, which he communicated 
to the Paris Academy in 1740 under the name of the 
" Loi de repos. " This principle was more fully dis- 
cussed by Euler in 1751 in the Proceedings of the 
Berlin Academy. If we cause infinitely small displace- 
TheLoide ments in any system, we produce a sum of virtual mo- 
repos - ments Pp + P'p' + P"p" +...., which only reduces 

to zero in the case of equilibrium. This sum is the 
work corresponding to the displacements, or since for 
infinitely small displacements it is itself infinitely small, 
the corresponding element of work. If the displace- 
ments are continuously increased till a finite displace- 
ment is produced, the elements of the work will, by 
summation, produce a finite amount of work. So, if we 



THE PRINCIPLES OF STATICS. 69 

start from anv given initial configuration of the system statement 

of the pria- 

and pass to any given final configuration, a certain cipie. 
amount of work will have to be done. Now Maupertuis 
observed that the work done when a final configura- 
tion is reached which is a configuration of equilibrium, 
is generally a maximum or a minimum ; that is, if we 
carry the system through the configuration of equilib- 
rium the work done is previously and subsequently 
less or previously and subsequently greater than at the 
configuration of equilibrium itself. For the configura- 
tion of equilibrium 

Pp + rp' + /"/ + = 

that is, the element of the work or the differential (more 
correctly the variation) of the work is equal to zero. 
If the differential of a function can be put equal to 
zero, the function has generally a maximum or mini- 
mum value. 

16. We can produce a very clear representation to Graphical 
the eye of the import of Maupertuis's principle. of the 

We imagine the forces of a system replaced by principle, 
sje's pulleys with the weight Q 2. We suppose 
that each point of the system is restricted to movement 
on a certain curve and that the motion is such that 

a one point occupies a definite position on its curve 
all the other points assume uniquely determined po- 
sitions on their respective curves. The simple ma- 
chines are as a rule systems of this kind. Now, while 
imparting displacements to the system, we may cany 
a vertical sheet of white paper horizontally over the 

ght Q 2, while this is ascending and descending 
on a vertical line, so that a pencil which it carries shall 
describe a curve upon the paper (Fig. 55). When the 
pencil stands at the points a, c, dot the curve, there are, 



7o 



THE SCIENCE OF MECHANICS. 



interpreta- we see, adjacent positions in the system of points at 

diagram, which the weight Q/2 will stand higher or lower than in 

the configuration given. The weight will then, if the 

system be left to itself, pass into this lower position and 




F>8- 55- 

displace the system with it. Accordingly, under condi- 
tions of this kind, equilibrium does not subsist. If 
the pencil stands at e, then there exist only adjacent 
configurations for which the weight Q/2 stands higher. 
But of itself the system will not pass into the last- 
named configurations. On the contrary, every dis- 
placement in such a direction, will, by virtue of the 
tendency of the weight to move downwards, be re- 
versed. Stable equilibrium, therefore, is the condition 
Stable equi- that corresponds to the loivest position of the weight or to 

librium. <' Tr , ., 

a maximum of work done in the system. It the pencil 
stands at b, we see that every appreciable displace- 
ment brings the weight Q/2 lower, and that the weight 
therefore will continue the displacement begun. But, 
assuming infinitely small displacements, the pencil 
moves in the horizontal tangent at b, in which event 
the weight cannot descend. Therefore, unstable equi- 
Unstabie librium is the state that corresponds to the highest position 
equilibrium ^ ^ w eight Q/2, or to a minimum of work done in the 
system. It will be noted, however, that conversely 



THE PRIXCIPLES OF STA TICS. 71 

everv case of equilibrium is not the correspondent of 
a maximum or a minimum of work performed. If the 
pencil is at/, at a point of horizontal contrary flexure, 
the weight in the case of infinitely small displace- 
ments neither rises nor falls. Equilibrium exists, al- 
though the work done is neither a maximum nor a 
minimum. The equilibrium of this case is the so- 
called mixed equilibrium * : for some disturbances it is Mixed equi- 

* librium. 

stable, for others unstable. Nothing prevents us from 
regarding mixed equilibrium as belonging to the un- 
stable class. When the pencil stands at g, where the 
curve runs along horizontally a finite distance, equi- 
librium likewise exists. Any small displacement, in 
the configuration in question, is neither continued nor 
reversed. This kind of equilibrium, to which likewise 
neither a maximum nor a minimum corresponds, is 
termed \neutral or] indifferent. If the curve described Neutral 
by Q/2 has a cusp pointing upwards, this indicates a eqmi num 
minimum of work done but no equilibrium (not even 
unstable equilibrium). To a cusp pointing downwards 
a maximum and stable equilibrium correspond. In the 
last named case of equilibrium the sum of the virtual 
moments is not equal to zero, but is negative. 

17. In the reasoning just presented, we have as- Thepreced- 

11 , r . t > n 8 iHustra- 

sumed that the motion of a point of a system on one t i on applied 
curve determines the motion of all the other points of t 6moredif- 
the system on their respective curves. The movability 
of the system becomes multiplex, however, when each 
point is displaceable on a surface, in a manner such 
that the position of one point on its surface determines 

* This term is not used in English, because our writers hold that no 
equilibrium is conceivable which is not stable or neutral for some possible 
displacements. Hence what is called mixed equilibrium in the text is called 
unstable equilibrium by English writers, who deny the existence of equilibrium 
unstable in every respect. Trans. 



72 . THE SCIENCE OF MECHANICS. 

uniquely the position of all the other points on their 
surfaces. In this case, we are not permitted to consider 
the curve described by Q/2, but are obliged to picture 
to ourselves a surface described by Q/2. If, to go a 
step further, each point is n^ovable throughout a space, 
we can no longer represent to ourselves in a purely geo- 
metrical manner the circumstances of the motion, by 
means of the locus of Q/2. In a correspondingly higher 
degree is this the case when the position of one of the 
points of the system does not determine conjointly all 
the other positions, but the character of the system's 
motion is more multiplex still. In all these cases, how- 
ever, the curve described by Q/2 (Fig. 55) can serve 
us as a symbol of the phenomena to be considered. In 
these cases also we rediscover the Maupertuisian pro- 
positions. 
Further ex- We have also supposed, in our considerations up to 
the same this point, that constant forces, forces independent of 
the position of the points of the system, are the forces 
that act in the system. If we assume that the forces 
do depend on the position of the points of the system 
(but not on the time), we are no longer able to conduct 
our operations with simple pulleys, but 
must devise apparatus the force active in 
which, still exerted by Q/2, varies with the 
displacement : the ideas we have reached, 
however, still obtain. The depth of the 
descent of the weight Q/2 is in every case 
the measure of the work performed, which 
is always the same in the same configura- 
Fi g- 56. tion of the system and is independent of 
the path of transference. A contrivance which would 
develop by means of a constant weight a force varying 
with the displacement, would be, for example, a wheel 




THE PRINCIPLES OF STATICS. 73 

and axle (Fig. 56; with a non-circular wheel. It would 
not repay the trouble, however, to enter into the de- 
tails of the reasoning indicated in this case, since we 
perceive at a glance its feasibility. 

18. If we know the relation that subsists between The prin- 

ciple of 
the work done and the so-called vis viva of a sys- Counivron. 

tern, a relation established in dynamics, we arrive 

easily at the principle communicated by Courtivrox in 

1749 to the Paris Academy, which is this: For the 

configuration of , , equilibrium, at which the 

work done is a max ! mum the vis viva of the system, 
minimum 

in motion, is also a max | mum J n its transit through 
minimum 

these configurations. 

19. A heavy, homogeneous triaxial ellipsoid resting ninstradon 
on a horizontal plane is admirably adapted to illustrate u ^|^ s u ^ 
the various classes of equilibrium. When the ellip- 
soid rests on the extremity of its smallest axis, it is in 

stable equilibrium, for any displacement it may suffer 
elevates its centre of gravity. If it rest on its longest 
axis, it is in unstable equilib- 
rium. If the ellipsoid stand on 
its mean axis, its equilibrium is 
mixed. A homogeneous sphere 
or a homogeneous right cylin- 
der on a horizontal plane illus- Fi e- 57. 
trates the case of indifferent equilibrium. In Fig. 57 
we have represented the paths of the centre of gravity 
of a cube rolling on a horizontal plane about one of its 
edges. The position a of the centre of gravity is the 
position of stable equilibrium, the position b, the posi- 
tion of unstable equilibrium. 




74 THE SCIENCE OF MECHANICS. 

The eaten- 20. We will now consider an example which at 

ary. 

first sight appears very complicated but is elucidated 
at once by the principle of virtual displacements. John 
and James Bernoulli, on the occasion of a conversa- 
tion on mathematical topics during a walk in Basel, 
lighted on the question of what form a chain would 
take that was freely suspended and fastened at both 
ends. They soon and easily agreed in the view that 
the chain would assume that form of equilibrium at 
which its centre of gravity lay in the lowest possible 
position. As a matter of fact we really do perceive 
that equilibrium subsists when all the links of the chain 
have sunk as low as possible, when none can sink lower 
without raising in consequence of the connections of 
the system an equivalent mass equally high or higher. 
When the centre of gravity has sunk as low as it pos- 
sibly can sink, when all has happened that can happen, 
stable equilibrium exists. The physical part of the 
problem is disposed of by this consideration. The de- 
termination of the curve that has the lowest centre of 
gravity for a given length between the two points A, 
B, is simply a mathematical problem. (See Fig. 58.) 
Theprinci- 21. Collecting all that has been presented, we see, 
piytherec- that there is contained in the principle of virtual dis- 

ognition of , . , , . . c r . 

a fact. placements simply the recognition of a fact that was 
instinctively familiar to us long previously, only that 
we had not apprehended it so precisely and clearly. 
This fact consists in the circumstance that heavy 
bodies, of themselves, move only downwards. If sev- 
eral such bodies be joined together so that they can 
suffer no displacement independently of each other, 
they will then move only in the event that some heavy 
mass is on the whole able to descend, or as the prin- 
ciple, with a more perfect adaptation of our ideas to 






THE PRIXCIPLES OF STATICS. 



75 




Fig. 58. 



76 THE SCIENCE OF MECHANICS. 

what this the facts, more exactly expresses it, only in the event 
that work can be performed. If, extending the notion 
of force, we transfer the principle to forces other than 
those due to gravity, the recognition is again con- 
tained therein of the fact that the natural occurrences 
in question take place, of themselves, only in a definite 
sense and not in the opposite sense. Just as heavy 
bodies descend downwards, so differences of tempera- 
ture and electrical potential cannot increase of their 
own accord but only diminish, and so on. If occur- 
rences of this kind be so connected that they can take 
place only in the contrary sense, the principle then es- 
tablishes, more precisely than our instinctive appre- 
hension could do this, the factor work as determinative 
and decisive of the direction of the occurrences. The 
equilibrium equation of the principle may be reduced 
in every case to the trivial statement, that when noth- 
ing can happen nothing does happen. 
Theprin- 22. It is important to obtain clearly the perception, 

Hghtof that we have to deal, in the case of all principles, 
view. merely with the ascertainment and establishment of a 

fact. If we neglect this, we shall always be sensible 
of some deficiency and will seek a verification of the 
principle, that is not to be found. Jacobi states in his 
Lectures on Dynamics that Gauss once remarked that 
Lagrange's equations of motion had not been proved, 
but only historically enunciated. And this view really 
seems to us to be the correct one in regard to the prin- 
ciple of virtual displacements. 
The differ- The task of the early inquirers, who lay the foun- 

ent tasks of . . . . r . . . . . , 

early and of dations of any department of investigation, is entirely 

fnqufrers in different from that of those who follow. It is the busi- 

ment. epa ness of the former to seek out and to establish the 

facts of most cardinal importance only; and, as history 



THE PRIXCIPLES OF STATICS. 77 

teaches, more brains are required for this than is gen- 
erally supposed. When the most important facts are 
once furnished, we are then placed in a position to 
work them out deductively and logically by the meth- 
ods of mathematical physics; we can then organise the 
department of inquiry in question, and show that in the 
acceptance of some one fact a whole series of others is 
included which were not to be immediately discerned 
in the first. The one task is as important as the other. 
We should not however confound the one with the 
other. We cannot prove by mathematics that nature 
must be exactly what it is. But we can prove, that 
one set of observed properties determines conjointly 
another set which often are not directly manifest. 

Let it be remarked in conclusion, that the princi- Every gen- 
ple of virtual displacements, like every general prin- pie brings 
ciple, brings with it, by the insight which it furnishes, niusion- 
disillusionment as well as elucidation. It brings with well as eiu- 

j. 1.. . , , ... cidation. 

it disillusionment to the extent that we recognise in it 
facts which were long before known and even instinct- 
ively perceived, our present recognition being simply 
more distinct and more definite ; and elucidation, in 
that it enables us to see everywhere throughout the 
most complicated relations the same simple facts. 

v. 

RETROSPECT OF THE DEVELOPMENT OF STATICS. 

i. Having passed successively in review the prin- Review of 
ciples of statics, we are now in a position to take a whole. 
brief supplementary survey of the development of the 
principles of the science as a whole. This development, 
falling as it does in the earliest period of mechanics, 
the period which begins in Grecian antiquity and 



78 THE SCIENCE OF MECHANICS. 

reaches its close at the time when Galileo and his 
younger contemporaries were inaugurating modern me- 
chanics, illustrates in an excellent manner the pro- 
cess of the formation of science generally. All con- 
ceptions, all methods are here found in their simplest 
form, and as it were in their infancy. These beginnings 
The origin point unmistakably to their origin in the experiences of 

of science 

the manual arts. To the necessity of putting these ex- 
periences into communicable form and of disseminating 
them beyond the confines of class and craft, science 
owes its origin. The collector of experiences of this 
kind, who seeks to preserve them in written form, finds 
before him many different, or at least supposably differ- 
ent, experiences. His position is one that enables him 
to review these experiences more frequently, more vari- 
ously, and more impartially than the individual work- 
ingman, who is always limited to a narrow province. 
The facts and their dependent rules are brought into 
closer temporal and spatial proximity in his mind and 
writings, and thus acquire the opportunity of revealing 
Theecono- their relationship, their connection, and their gradual 

my of com- ... . . . ,_,. . 

munication. transition the one into the other. lhe desire to sim- 
plify and abridge the labor of communication supplies 
a further impulse in the same direction. Thus, from 
economical reasons, in such circumstances, great num- 
bers of facts and the rules that spring from them are 
condensed into a system and comprehended in a single 
expression. 
The gene- 2. A collector of this character has, moreover, op- 

ter of prin- portunity to take note of some new aspect of the facts 

ciples. 

before him of some aspect which former observers 
had not considered. A rule, reached by the observation 
of facts, cannot possibly embrace the entire fact, in all 
its infinite wealth, in all its inexhaustible manifoldness; 



THE PRINCIPLES OF STATICS. 79 

on the contrary, it can furnish only a rough outline of 
the fact, one-sidedly emphasising the feature that is of 
importance for the given technical (or scientific) aim in 
view. What aspects of a fact are taken notice of, will 
consequentlv depend upon circumstances, or even on Their form 

" . > n many as- 

the caprice of the observer. Hence there is alwavs op- pects, acci- 

. . t dental. 

portunity for the discovery of new aspects of the fact, 
which will lead to the establishment of new rules of 
equal validity with, or superior to, the old. So, for in- 
stance, the weights and the lengths of the lever-arms 
were regarded at first, by Archimedes, as the conditions 
that determined equilibrium. Afterwards, by Da Vinci 
and Ubaldi the weights and the perpendicular distances 
from the axis of the lines of force were recognised as 
the determinative conditions. Still later, by Galileo, 
the weights and the amounts of their displacements, 
and finally by Yarignon the weights and the directions 
of the pulls with respect to the axis were taken as the 
elements of equilibrium, and the enunciation of the 
rules modified accordingly. 

3. Whoever makes a new observation of this kind, om-iiaoii- 
and establishes such a new rule, knows, of course, our in the men- 

...... . talrecon- 

liability to error in attempting mentally to represent stmction of 
the fact, whether by concrete images or in abstract con- 
ceptions, which we must do in order to have the mental 
model we have constructed always at hand as a substi- 
tute for the fact when the latter is partly or wholly in- 
accessible. The circumstances, indeed, to which we 
have to attend, are accompanied by so many other, 
collateral circumstances, that it is frequently difficult 
to single out and consider those that are essential to the 
purpose in view. Just think how the facts of friction, 
the rigidity of ropes and cords, and like conditions in 
machines, obscure and obliterate the pure outlines of 



80 THE SCIENCE OF MECHANICS. 

Thisiiabil- the main facts. No wonder, therefore, that the discov- 

lty impels 

us to seek erer or verifier of a new rule, urged by mistrust of him- 

after proofs 

of all new self, seeks after a proof of the rule whose validity he 

rules. . . * 

believes he has discerned. The discoverer or verifier 
does not at the outset fully trust in the rule ; or, it may 
be, he is confident only of a part of it. So, Archimedes, 
for example, doubted whether the effect of the action 
of weights on a lever was proportional to the lengths of 
the lever-arms, but he accepted without hesitation the 
fact of their influence in some way. Daniel Bernoulli 
does not question the influence of the direction of a 
force generally, but only the form of its influence. As 
a matter of fact, it is far easier to observe that a circum- 
stance has influence in a given case, than to determine 
what influence it has. In the latter inquiry we are in 
much greater degree liable to error. The attitude of the 
investigators is therefore perfectly natural and defens- 
ible. 
The natural The proof of the correctness of a new rule can be 
proof. attained by the repeated application of it, the frequent 
comparison of it with experience, the putting of it to 
the test under the most diverse circumstances. This 
process would, in the natural course of events, get car- 
ried out in time. The discoverer, however, hastens to 
reach his goal more quickly. He compares the results 
that flow from his rule with all the experiences with 
which he is familiar, with all older rules, repeatedly 
tested in times gone by, and watches to see if he do 
not light on contradictions. In this procedure, the I 
greatest credit is, as it should be, conceded to the oldest ! 
and most familiar experiences, the most thoroughly j 
tested rules. Our instinctive experiences, those gen- 
eralisations that are made involuntarily, by the irresist- 
ible force of the innumerable facts that press in upon 



THE PRIXCIPLES OF STATICS. 8r. 

us, enjoy a peculiar authority; and this is perfectly 
warranted by the consideration that it is precisely the 
elimination of subjective caprice and of individual er- 
ror that is the object aimed at. 

In this manner Archimedes proves his law of the illustration 
lever, Stevinus his law of inclined pressure, Daniel ceding re- 
Bernoulli the parallelogram of forces, Lagrange the 
principle of virtual displacements. Galileo alone is 
perfectly aware, with respect to the last-mentioned 
principle, that his new observation and perception are 
of equal rank with every former one that it is derived 
from the same source of experience. He attempts no 
demonstration. Archimedes, in his proof of the prin- 
ciple of the lever, uses facts concerning the centre of 
gravity, which he had probably proved by means of the 
very principle now in question ; yet we may suppose 
that these facts were otherwise so familiar, as to be un- 
questioned, so familiar indeed, that it may be doubted 
whether he remarked that he had employed them in 
demonstrating the principle of the lever. The instinc- 
tive elements embraced in the views of Archimedes and 
Stevinus have been discussed at length in the proper 
place. 

4. It is quite in order, on the making of a new dis-Theposi- 

. tionthatad- 

covery, to resort to all proper means to bring the new vanced sci- 
rule to the test. When, however, after the lapse of a occupy, 
reasonable period of time, it has been sufficiently often 
subjected to direct testing, it becomes science to recog- 
nise that any other proof than that has become quite 
needless ; that there is no sense in considering a rule 
as the better established for being founded on others 
that have been reached by the very same method of 
observation, only earlier ; that one well-considered and 
tested observation is as good as another. To-day, we 



82 THE SCIENCE OF MECHANICS. 

should regard the principles of the lever, of statical 
moments, of the inclined plane, of virtual displace- 
ments, and of the parallelogram of forces as discovered 
by equivalent observations. It is of no importance now, 
that some of these discoveries were made directly, while 
others were reached by roundabout ways and as de- 
pendent upon other observations. It is more in keep- 
ing, furthermore, with the economy of thought and with 
insight bet- the aesthetics of science, directly to recognise a principle 

ter than ar- . J d JT- r 

tificiaidem- (say that of the statical moments) as the key to the un- 

onstration. J J 

derstandmg of all the facts of a department, and really 
see how it pervades all those facts, rather than to hold 
ourselves obliged first to make a clumsy and lame de- 
duction of it from unobvious propositions that involve 
the same principle but that happen to have become 
earlier familiar to us. This process science and the in- 
dividual (in historical study) may go through once for 
all. But having done so both are free to adopt a more 
convenient point of view. 
Themis- 5. In fact, this mania for demonstration in science 

mania for results in a rigor that is false and mistaken. Some pro- 
tion. * positions are held to be possessed of more certainty 
than others and even regarded as their necessary and 
incontestable foundation ; whereas actually no higher, 
or perhaps not even so high, a degree of certainty at- 
taches to them. Even the rendering clear of the de- , 
gree of certainty which exact science aims at, is not at- j 
tained here. Examples of such mistaken rigor are to 
be found in almost every text-book. The deductions' 
of Archimedes, not considering their historical value, 
are infected with this erroneous rigor. But the mosti 
conspicuous example of all is furnished by Daniel Ber- 
noulli's deduction of the parallelogram of forces {Com-. 
ment. Acad. Petrop. T. I.). 



THE PRINCIPLES OF STATICS. 83 

6. As already seen, instinctive knowledge enjoys The char- 

. acter of in- 

our exceptional confidence. No longer knowing how stincuve 

. ... ....... knowledge. 

we have acquired it, we cannot criticise the logic by 
which it was inferred. We have personally contributed 
nothing to its production. It confronts us with a force 
and irresistibleness foreign to the products of volun- 
tary reflective experience. It appears to us as some- 
thing free from subjectivity, and extraneous to us, al- 
though we have it constantly at hand so that it is more 
ours than are the individual facts of nature. 

All this has often led men to attribute knowledge of its anthor- 
this kind to an entirely different source, namely, to view lu'teiy su- 

.... ,. . preme. 

it as existing a priori in us (previous to all experience). 
That this opinion is untenable was fully explained in 
our discussion of the achievements of Stevinus. Yet 
even the authority of instinctive knowledge, however 
important it may be for actual processes of develop- 
ment, must ultimately give place to that of a clearly and 
deliberately observed principle. Instinctive knowledge 
is, after all, only experimental knowledge, and as such 
is liable, we have seen, to prove itself utterly insuffi- 
cient and powerless, when some new region of expe- 
rience is suddenly opened up. 

7. The true relation and connection of the different The tmere- 
principles is the historical one. The one extends farther principles 
in this domain, the other farther in that. Notwith- cai one. 

I standing that some one principle, say the principle of 
! virtual displacements, may control with facility a 
greater number of cases than other principles, still 
no assurance can be given that it will always maintain 
its supremacy and will not be outstripped by some new 
principle. All principles single out, more or less arbi- 
trarily, now this aspect now that aspect of the same 
facts, and contain an abstract summarised rule for the 



84 THE SCIENCE OF MECHANICS. 

refigurement of the facts in thought. We can never 
assert that this process has been definitively completed. 
Whosoever holds to this opinion, will not stand in the 
way of the advancement of science. 
Conception 8. Let us, in conclusion, direct our attention for a 

of force in .... 

statics. moment to the conception of force in statics. Force is 
any circumstance of which the consequence is motion. 
Several circumstances of this kind, however, each single 
one of which determines motion, may be so conjoined 
that in the result there shall be no motion. Now stat- 
ics investigates what this mode of conjunction, in gen- 
eral terms, is. Statics does not further concern itself 
about the particular character of the motion condi- 
tioned by the forces. The circumstances determinative 
of motion that are best known to us, are our own vo- i 

The origin litional acts our innervations. In the motions which ; 

tionof we ourselves determine, as well as in those to which j 
we are forced by external circumstances, we are always 
sensible of a pressure. Thence arises our habit of rep- 
resenting all circumstances determinative of motion as 
something akin to volitional acts as pressures. The 
attempts we make to set aside this conception, as sub- 
jective, animistic, and unscientific, fail invariably. It 
cannot profit us, surely, to do violence to our own nat- 
ural-born thoughts and to doom ourselves, in that re- 
gard, to voluntary mental penury. We shall subse- 
quently have occasion to observe, that the conception 
referred to also plays a part in the foundation of dy- 
namics. 

We are able, in a great many cases, to replace the 
circumstances determinative of motion, which occur in 
nature, by our innervations, and thus to reach the idea 
of a gradation of the intensity of forces. But in the esti- 
mation of this intensity we are thrown entirely on the 



pressure. 



THE PRINCIPLES OF STATICS. 85 

resources of our memory, and are also unable to com-Thecom- 

. . -iii mon cnar- 

municate our sensations. Since it is possible, how- acter of all 

. forces. 

ever, to represent every condition that determines 
motion by a weight, we arrive at the perception that 
all circumstances determinative of motion (all forces) 
are alike in character and may be replaced and meas- 
ured by quantities that stand for. weight. The meas- 
urable weight serves us, as a certain, convenient, and 
communicable index, in mechanical researches, just as 
the thermometer in thermal researches is an exacter 
substitute for our perceptions of heat. As has pre- The idea of 

-l. 11 -j .-motion an 

viouslv been remarked, statics cannot wholly rid itseli auxiliary 

- ,,,, . concept in 

of all knowledge of phenomena of motion. This par- statics, 
ticularly appears in the determination of the direction 
of a force by the direction of the motion which it would 
produce if it acted alone. By the point of application 
of a force we mean that point of a body whose motion 
is still determined by the force when the point is freed 
from its connections with the other parts of the body. 

Force accordingly is any circumstance that de-Thegene- 

J J , ral attri- 

termines motion ; and its attributes may be stated asbutesof 

. . . r force. 

follows. The direction of the force is the direction of 
motion which is determined by that force, alone. The 
point of application is that point whose motion is de- 
termined independently of its connections with the 
system. The magnitude of the force is that weight 
which, acting (say, on a string) in the direction deter- 
mined, and applied at the point in question, determines 
the same motion or maintains the same equilibrium. 
The other circumstances that modify the determination 
of a motion, but by themselves alone are unable to pro- 
duce it, such as virtual displacements, the arms of 
levers, and so forth, may be termed collateral condi- 
tions determinative of motion and equilibrium. 



85 THE SCIENCE OF MECHANICS. 



THE PRINCIPLES OF STATICS IN THEIR APPLICATION TO 
FLUIDS. 

Noessen- I. The consideration of fluids has not supplied stat- 

p'oints^r ics with many essentially new points of view, yet nu- 
volvelfin merous applications and confirmations of the principles 
issu J ect - already known have resulted therefrom, and physical 
experience has been greatly enriched by the investiga- 
tions of this domain. We shall devote, therefore, a few 
pages to this subject. 

2. To Archimedes also belongs the honor of found- 
ing the domain of the statics of liquids. To him we 
owe the well-known proposition concerning the buoy- 
ancy, or loss of weight, of bodies immersed in liquids, 
of the discovery of which Vitruvius, De Architecture!,, 
Lib. IX, gives the following account : 
Vitruvius's " Though Archimedes discovered many curious 
Archime- "matters that evince great intelligence, that which I am 
covery. 1S " " about to mention is the most extraordinary. Hiero, 
"when he obtained the regal power in Syracuse, hav- 
ing, on the fortunate turn of his affairs, decreed a 
"votive crown of gold to be placed in a certain temple 
"to the immortal gods, commanded it to be made of 
"great value, and assigned for this purpose an appro- 
"priate weight of the metal to the manufacturer. The 
"latter, in due time, presented the work to the king, 
"beautifully wrought ; and the weight appeared to cor- 
" respond with that of the gold which had been as- 
signed for it. 

"But a report having been circulated, that some of 
"the gold had been abstracted, and that the deficiency 



THE PRIXCIPLES OF STATICS. 87 

"thus caused had been supplied by silver, Hiero was The ac- 
" indignant at the fraud, and, unacquainted with the tmvius. 
"method by which the theft might be detected, re- 
quested Archimedes would undertake to give it his 
"attention. Charged with this commission, he by 
"chance went to a bath, and on jumping into the tub, 
"perceived that, just in the proportion that his body 
"became immersed, in the same proportion the water 
"ran out of the vessel. Whence, catching at the 
" method to be adopted for the solution of the proposi- 
tion, he immediately followed it up, leapt out of the 
"vessel in joy, and returning home naked, cried out 
"with a loud voice that he had found that of which he 
"was in search, for he continued exclaiming, in Greek, 
jt)xa, evprjna, (I have found it, I have found it !)" 
3. The observation which led Archimedes to his statement 

. . . of the Ar- 

proposition, was accordingly this, that a body 1m- chimedean 

. . proposition 

mersed in water must raise an equivalent quantity of 
water ; exactly as if the body lay on one pan of a balance 
and the water on the other. This conception, which 
at the present day is still the most natural and the 
most direct, also appears in Archimedes's treatises On 
Floating Bodies, which unfortunately have not been 
completely preserved but have in part been restored 
by F. Commandinus. 

The assumption from which Archimedes starts 
reads thus : 

" It is assumed as the essential property of a liquid The Archi- 

.,,.,. . r . . . . . . medeanas- 

tnat in all uniform and continuous positions of its parts sumption, 
the portion that suffers the lesser pressure is forced 
upwards by that which suffers the greater pressure. 
But each part of the liquid suffers pressure from the 
portion perpendicularly above it if the latter be sinking 
or suffer pressure from another portion." 



88 



THE SCIENCE OF MECHANICS. 



Analysis of 
the princi- 
ple. 




Fig. 59- 



Archimedes now, to present the matter briefly, 
conceives the entire spherical earth as fluid in consti- 
tution, and cuts out of it pyramids the vertices of 
which lie at the centre (Fig. 59). All these pyramids 

must, in the case of equilib- 
rium, have the same weight, 
and the similarly situated 
parts of the same must all 
suffer the same pressure. 
If. we plunge a body a of 
the same specific gravity as 
water into one of the pyra- 
mids, the body will com- 
pletely submerge, and, in 
the case of equilibrium, will supply by its weight the 
pressure of the displaced water. The body b, of less 
specific gravity, can sink, without disturbance of equi- 
librium, only to the point at which the water beneath 
it suffers the same pressure from the weight of the 
body as it would if the body were taken out and the 
submerged portion replaced by water. The body c, 
of a greater specific gravity, sinks as deep as it possibly 
can. That its weight is lessened in the water by an 
amount equal to the weight of the water displaced, 
will be manifest if we imagine the body joined to 
another of less specific gravity so that a third body is 
formed having the same specific gravity as water, 
which just completely submerges, 
rhe state of 4. When in the sixteenth century the study of the 
mine six- works of Archimedes was again taken up, scarcely the 
tury. principles of his researches were understood. The 

complete comprehension of his deductions was at that 
time impossible. 

Stevinus rediscovered by a method of his own the 



THE PRIXCIPLES OF STATICS. 



89 



most important principles of hydrostatics and the de- The teat- 

. .... . , t eriesofSte- 

ductions therefrom. It was principally two ideas fromvinus. 
which Stevinus derived his fruitful conclusions. The 
one is quite similar to that relating to the endless 
chain. The other consists in the assumption that the 
solidification of a fluid in equilibrium does not disturb 
its equilibrium. 

Stevinus first lays down this principle. Any given The first 

J ... . . fundamen- 

mass of water A (Fig. 60), immersed in water, is intaiprinci- 
equilibrium in all its parts. If A 



A 



Fig. 60. 



were not supported by the sur- 
rounding water but should, let us 
say, descend, then the portion of 
water taking the place of A and 
placed thus in the same circum- 
stances, would, on the same as- 
sumption, also have to descend. 
This assumption leads, therefore, to the establishment 
of a perpetual motion, which is contrary to our ex- 
perience and to our instinctive knowledge of things. 

Water immersed in water loses accordingly its The second 

fundamen- 

whole weight. If, now, we imagine the surface of the tai prind- 
submerged water solidified, the vessel formed by this 
surface, the vas superficiarium as Stevinus calls it, will 
still be subjected to the same circumstances of pres- 
sure. If empty, the vessel so formed will suffer an 
upward pressure in the liquid equal to the weight of the 
water displaced. If we fill the solidified surface with 
some other substance of any specific gravity we may 
choose, it will be plain that the diminution of the 
weight of the body will be equal to the weight of the 
fluid displaced on immersion. 

In a rectangular, vertically placed parallelepipedal 
vessel filled with a liquid, the pressure on the. horizontal 



go 



THE SCIENCE OF MECHANICS 



stevinus's base is equal to the weight of the liquid. The pressure 

deductions. . 

is equal, also, for all parts of the bottom of the same 
area. When now Stevinus imagines portions of the 
liquid to be cut out and replaced by rigid immersed 
bodies of the same specific gravity, or, what is the 
same thing, imagines parts of the liquid to become so- 
lidified, the relations of pressure in the vessel will not 
be altered by the procedure. But we easily obtain in 
this way a clear view of the law that the pressure on 
the base of a vessel is independent of its form, as well 
as of the laws of pressure in communicating vessels, 
and so forth. 
Galileo, in 5. Galileo treats the equilibrium of liquids in com- 

the treat- . ^ n 

mentof this mumcatmg vessels and the problems connected there- 
subject, em- . 
ploys the with by the help of the principle of virtual displace- 

principle of J r if f 

virtual dis- _ ments. NN (Fig. 61) being the 



placements 



B 



A 



S 



common level of a liquid in equilib- 
rium in two communicating vessels, 



Fig. 61. 



CT >N Galileo explains the equilibrium 
here presented by observing that in 
the case of any disturbance the dis- 
placements of the columns are to 
each other in the inverse proportion 
of the areas of the transverse sec- 
tions and of the weights of the columns that is, as 
with machines in equilibrium. But this is not quite cor- 
rect. The case does not exactly correspond to the 
cases of equilibrium investigated by Galileo in ma- 
chines, which present indifferent equilibrium. With 
liquids in communicating tubes every disturbance of the 
common level of the liquids produces an elevation of 
the centre of gravity. In the case represented in Fig. 
61, the centre of gravity 6" of the liquid displaced from 
the shaded space in A is elevated to S' , and we may 



THE PRIXCIPLES OF STATICS. 



9i 




Fig. 62. 



regard the rest of the liquid as not having been moved. 
Accordingly, in the case of equilibrium, the centre of 
gravity of the liquid lies at its lowest possible point. 

6. Pascal likewise employs the principle of virtual The same 

. principle 

displacements, but in a more correct manner, leaving made useof 
the weight of the liquid out of account and considering 
only the pressure at the surface. If we imagine two 
communicating vessels to be closed by pistons (Fig. 
62), and these pistons loaded with 
weights proportional to their surface- 
areas, equilibrium will obtain, because 
in consequence of the invariability of 
the volume of the liquid the displace- 
ments in every disturbance are in- 
versely proportional to the weights. 
For Pascal, accordingly, it follows, as a necessary con- 
sequence, from the principle of virtual displacements, 
that in the case of equilibrium every pressure on a su- 
perficial portion of a liquid is propagated with undi- 
minished effect to every other superficial portion, how- 
ever and in whatever position it be placed. No objec- 
tion is to be made to discovering the principle in this 
way. Yet we shall see later on that the more natural 
and satisfactory conception is to regard the principle as 
immediately given. 

7. We shall now, after this historical sketch, again Detailed 

, . r1 . . . .... . considera- 

examinethe most important cases of liquid equilibrium, tionof the 
and from such different points of view as may be con- 
venient. 

The fundamental property of liquids given us by 
experience consists in the flexure of their parts on the 
slightest application of pressure. Let us picture to our- 
selves an element of volume of a liquid, the gravity of 
which we disregard say a tiny cube. If the slightest 



92 THE SCIENCE OF MECHANICS. 

The funda- excess of pressure be exerted on one of the surfaces of 

mental / 

property of this cube, (which we now conceive, for the moment, 

liquids the _ , , . . . 

mobility of as a fixed geometrical locus, containing the fluid but 

their parts. . 

not of its substance) the liquid (supposed to have pre- 
viously been in equilibrium and at rest) will yield and 
pass out in all directions through the other five surfaces 
of the cube. A solid cube can stand a pressure on its 
upper and lower surfaces different in magnitude from 
that on its lateral surfaces ; or vice versa. A fluid cube, 
on the other hand, can retain its shape only if the same 
perpendicular pressure be exerted on all its sides. A 
similar train of reasoning is applicable to all polyhe- 
drons. In this conception, as thus geometrically eluci- 
dated, is contained nothing but the crude experience 
that the particles of a liquid yield to the slightest pres- 
sure, and that they retain this property also in the in- 
terior of the liquid when under a high pressure ; it 
being observable, for example, that under the condi- 
tions cited minute heavy bodies sink in fluids, and so on. 
A second With the mobility of their parts liquids combine 

the com- still another property, which we will now consider. Li- 
of their vol- quids suffer through pressure a diminution of volume 
which is proportional to the pressure exerted on unit 
of surface. Every alteration of pressure carries along 
with it a proportional alteration of volume and density. 
If the pressure diminish, the volume becomes greater, 
the density less. The volume of a liquid continues to 
diminish therefore on the pressure being increased, till 
the point is reached at which the elasticity generated 
within it equilibrates the increase of the pressure. 

8. The earlier inquirers, as for instance those of the 
Florentine Academy, were of the opinion that liquids 
were incompressible. In 1761, however, John Canton 
performed an experiment by which the compressibility 



THE PKIXCIPLES OF STATICS. 



93 



of water was demonstrated. A thermometer glass is The first 

demonstra- 

filled with water, boiled, and then sealed. (Fig. 63.) tionof ue 

. compressi- 

The liquid reaches to a. But since the space above a is biiity of 

. liquids. 

airless, the liquid supports no atmospheric pres- 
sure. If the sealed end be broken off, the liquid 
will sink to b. Only a portion, however, of this 
displacement is to be placed to the credit of the ^[b 
compression of the liquid by atmospheric pres- 
sure. For if we place the glass before breaking ^^ 
off the top under an air-pump and exhaust the ^^ 
chamber, the liquid will sink to c. This last phe- 
nomenon is due to the fact that the pressure that 
bears down on the exterior of the glass and diminishes 
its capacity, is now removed. On breaking off the top, 
this exterior pressure of the atmosphere is compensated 
for by the interior pressure then introduced, and an 
enlargement of the capacity of the glass again sets in. 
The portion cb, therefore, answers to the actual com- 
pression of the liquid by the pressure of the atmos- 
phere. 

The first to institute exact experiments on the com- The experi- 
ments of 
pressibility of water, was Oersted, who emploj^ed to oersted on 

this end a very ingenious method. A 
thermometer glass A (Fig. 64) is filled 
with boiled water and is inverted, with 
open mouth, into a vessel of mercury. 
Near it stands a manometer tube B filled 
with air and likewise inverted with open 
mouth in the mercury. The whole ap- 
paratus is then placed in a vessel filled 
with water, which is compressed by the 
aid of a pump. By this means the water 
in A is also compressed, and the filament of quicksilver 
which rises in the capillar)' tube of the thermometer- 




Fig. 64. 



94 THE SCIENCE OF MECHANICS. 

glass indicates this compression. The alteration of 
capacity which the glass A surfers in the present in- 
stance, is merely that arising from the pressing to- 
gether of its walls by forces which are equal on all sides. 
The experi- The most delicate experiments on this subject have 

ments of 

Grassi. been conducted by Grassi with an apparatus con- 
structed by Regnault, and computed with the assist- 
ance of Lame's correction-formulae. To give a tan- 
gible idea of the compressibility of water, we will remark 
that Grassi observed for boiled water at under an 
increase of one atmospheric pressure a diminution of 
the original volume amounting to 5 in 100,000 parts. 
If we imagine, accordingly, the vessel A to have the 
capacity of one litre (1000 ccm.), and affix to it a cap- 
illary tube of 1 sq. mm. cross-section, the quicksilver 
filament will ascend in it 5 cm. under a pressure of 
one atmosphere. 

Surface- 9. Surface-pressure, accordingly, induces a physical 

pressure in- . . . . . , , . . . . 

duces in alteration in a liquid (an alteration in density), which 
alteration can be detected by sufficiently delicate means even 
optical. We are always at liberty to think that por- 
tions of a liquid under a higher pressure are more dense, 
though it may be very slightly so, than parts under a 
less pressure. 
The impii- Let us imagine now, we have in a liquid (in the in- 

cations of . ...... r ., . . . 

this fact, tenor of which no forces act and the gravity of which 
we accordingly neglect) two portions subjected to un- 
equal pressures and contiguous to one another. The 
portion under the greater pressure, being denser, will 
expand, and press against the portion under the less 
pressure, until the forces of elasticity as lessened on the 
one side and increased on the other establish equilib- 
rium at the bounding surface and both portions are 
equally compressed. 



THE PRINCIPLES OF STATICS. 95 

If we endeavor, now, quantitatively to elucidate our The state- 
mental conception of these two facts, the easy mobility these impii- 
and the compressibility of the parts of a liquid, so that 
they will fit the most diverse classes of experience, 
we shall arrive at the following proposition : When 
equilibrium subsists in a liquid, in the interior of which 
no forces act and the gravity of which we neglect, the 
same equal pressure is exerted on each and every equal 
surface-element of that liquid however and wherever 
situated. The pressure, therefore, is the same at all 
points and is independent of direction. 

Special experiments in demonstration of this prin- 
ciple have, perhaps, never been instituted with the re- 
quisite degree of exactitude. But the proposition has 
by our experience of liquids been made very familiar, 
and readily explains it. 

10. If a liquid be enclosed in a vessel (Fig. 65) Preiimi- 

,.,. ..... . . nary re- 

which is supplied with a piston^, the cross-section marks to 

r 1 1 * n the discuss- 

01 which is unit in area, and with a piston B which ion of Pas- 

..... . cal'sdeduc- 

tor the time being is made station- m I m 4 tion- 

ary, and on the piston A a load p ^^WS^ ^ 

be placed, then the same pressure i g -~ r ? j^jm 
p, gravity neglected, will prevail W_- ~~ ^ r^3 
throughout all the parts of the vessel. J blt- -"^ : --'J 
The piston will penetrate inward and \a8 S^^ ~^r 
the walls of the vessel will continue ihjr 
to be deformed till the point is reached Fi e- 6 5- 

at which the elastic forces of the rigid and fluid bodies 
perfectly equilibrate one another. If then we imagine 
the piston B, which has the cross-section/, to be mov- 
able, a force f.p alone will keep it in equilibrium. 

Concerning Pascal's deduction of the proposition 
before discussed from the principle of virtual displace- 
ments, it is to be remarked that the conditions of dis- 



9 6 



THE SCIENCE OF MECHANICS. 



Criticism of placement which he perceived hinge wholly upon the 
duction. fact of the ready mobility of the parts and on the 
equality of the pressure throughout every portion of 
the liquid. If it were possible for a greater compression 
to take place in one part of a liquid than in another, 
the ratio of the displacements would be disturbed and 
Pascal's deduction would no longer be admissible. 
That the property of the equality of the pressure is a 
property given in experience, is a fact that cannot be 
escaped ; as we shall readily admit if we recall to mind 
that the same law that Pascal deduced for liquids also 
holds good for gases, where even approximately there 
can be no question of a constant volume. This latter 
fact does not afford any difficulty to our view ; but to 
that of Pascal it does. In the case of the lever also, be 
it incidentally remarked, the ratios of the virtual dis- 
placements are assured by the elastic forces of the 
lever-body, which do not permit of any great devia- 
tion from these relations. 
Thebehav- j j # We shall now consider the action of liquids un- 

lourofli- 

auids under der the influence of gravity. The upper surface of a 

the action .. . .. .. 

liquid in equilibrium is horizontal, 
NN (Fig. 66). This fact is at once 
rendered intelligible when we re- 
flect that every alteration of the sur- 
face in question elevates the centre 
of gravity of the liquid, and pushes 
the liquid mass resting in the shaded 
space beneath NN and having the centre of gravity S 
into the shaded space above NN having the centre of 
gravity S' . Which alteration, of course, is at once re- : 
versed by gravity. 

Let there be in equilibrium in a vessel a heavy 
liquid with a horizontal upper surface. We consider 



of gravity. 




Fig. 66. 



THE PRINCIPLES OE STATICS. 



97 



P+dp 



Fig. 67. 



(Fig. 67) a small rectangular parallelepipedon in the 
interior. The area of its horizontal base, we will say. is 
a, and the length of its vertical edges dh. The weight 
of this parallelepipedon is therefore adhs, where s is 
its specific gravity. If the paral- 
lelepipedon do not sink, this is 
possible only on the condition that 
a greater pressure is exerted on the 
lower surface by the fluid than on 
the upper. The pressures on the 
upper and lower surfaces we will 
respectively designate as ccp and a (/ -f- dp). Equi- 
librium obtains when ad/i.s = a dp or dpjdh = s, 
where h in the downward direction is reckoned as posi- 
tive. We see from this that for equal increments of // 
vertically downwards the pressure / must, correspond- 
ingly, also receive equal increments. So that p = 
q ; and if q, the pressure at the upper surface, 
which is usually the pressure of the atmosphere, be- 
comes = 0, we have, more simply, p = /is, that is, the 
pressure is proportional to the depth beneath the sur- 
face. If we imagine the liquid to be pouring into a ves- 
sel, and this condition of affairs uOi vd attained, every 
liquid particle will then sink until the compressed par- 
ticle beneath balances by the elasticity developed in it 
the weight of the particle above. 

From the view we have here presented it will be fur- 
ther apparent, that the increase of pressure in a liquid 
takes place solely in the direction in which gravity 
acts. Only at the lower surface, at the base, of the 
parallelepipedon, is an excess of elastic pressure on the 
part of the liquid beneath required to balance the 
weight of the parallelepipedon. Along the two sides of 
the vertical containing surfaces of the parallelepipedon, 



The con- 
ditions of 
equilibrium 
in liquids 
subjected 
to the a - 
tion of grav- 
ity. 



Different 
force-rela- 
tions exist 
only in the 
line of the 
action of 
gravity. 



98 THE SCIENCE OS MECHANICS. 

the liquid is in a state of equal compression, since no 
force acts in the vertical containing surfaces that would 
determine a greater compression on the one side than 
on the other. 
Level sur- If we picture to ourselves the totality of all the 

points of the liquid at which the same pressure / acts, 
we shall obtain a surface a so-called level surface. If 
we displace a particle in the direction of the action of 
gravity, it undergoes a change of pressure. If we dis- 
place it at right angles to the direction of the action of 
gravity, no alteration of pressure takes place. In the 
latter case it remains on the same level surface, and 
the element of the level surface, accordingly, stands at 
right angles to the direction of the force of gravity. 

Imagining the earth to be fluid and spherical, the 
level surfaces are concentric spheres, and the directions 
of the forces of gravity (the radii) stand at right angles 
to the elements of the spherical surfaces. Similar ob- 
servations are admissible if the liquid particles be acted 
on by other forces than gravity, magnetic forces, for 
example. 
Their func- The level surfaces afford, in a certain sense, a dia- ' 
thought, gram of the force-relations to which a fluid is subjected; 
a view further elaborated by analytical hydrostatics. 

12. The increase of the pressure with the depth be- j 
low the surface of a heavy liquid may be illustrated by 
a series of experiments which we chiefly owe to Pas- 
cal. These experiments also well illustrate the fact, 
that the pressure is independent of the direction. In 
Fig. 68, i, is an empty glass tube g ground off at the 
bottom and closed by a metal disc pp, to which ai 
string is attached, and the whole plunged into a vessel 
of water. When immersed to a sufficient depth wt 
may let the string go, without the metal disc, which ii 



THE PRINCIPLES OF STATICS. 



99 




supported by the pressure of the liquid, falling. In 2, 
the metal disc is replaced by a tiny column of mer- 
cury. If (3) we dip an open siphon tube filled with 
mercury into the water, we shall 
see the mercury, in consequence 
of the pressure at a, rise into 
the longer arm. In 4, we see a 
tube, at the lower extremity of 
which a leather bag filled with 
mercury is tied : continued im- 
mersion forces the mercury 
higher and higher into the tube. 
In 5, a piece of wood // is driven 
by the pressure of the water into 
the small arm of an empty siphon [ 
tube. In 6, a piece of wood H 
immersed in mercury adheres to 
the bottom of the vessel, and is 
pressed firmly against it for as 
long a time as the mercury is , 
kept from working its way un- 
derneath it. 

13. Once we have made quite 
clear to ourselves that the pres- 
sure in the interior of a heavy 
liquid increases proportionally to 
the depth below the surface, the 
law that the pressure at the base 
of a vessel is independent of its 
form will be readily perceived. 
The pressure increases as we de- 
scend at an equal rate, whether the vessel (Fig. 69) 
has the form abed or ebef. In both cases the walls 
of the vessel where they meet the liquid, go on deforming 



Pascal's ex- 
periments 
on the 
pressure of 
liquids. 




THE SCIENCE OF MECHANICS. 



Elucida- 
tion of this 
fact. 




Fig. 69. 



The princi- 
ple of vir- 
tual dis- 
placements 
applied to 
the consid- 
eration of 
problems of 
this class. 



till the point is reached at which they equilibrate by the 
elasticity developed in them the pressure exerted by the 
fluid, that is, take the place as regards pressure of the 

fluid adjoining. This fact is 
a direct justification of Ste- 
vinus's fiction of the solidi- 
fied fluid supplying the place 
of the walls of the vessel. 
The pressure on the base 
always remains P = A /is, 
where A denotes the area of the base, h the depth of 
the horizontal plane base below the level, and s the 
specific gravity of the liquid. 

The fact that, the walls of the vessel being neg- 
lected, the vessels 1, 2, 3 of Fig. 70 of equal base- 
area and equal pressure-height weigh differently in the 

balance, of course 
in no wise con- 
tradicts the laws 
of pressure men- 
tioned. If we take 
Fig. 70- into account the 

lateral pressure, we shall see that in the case of 1 we 
have left an extra component downwards, and in the 
case of 3 an extra component upwards, so that on the 
whole the resultant superficial pressure is always equal 
to the weight. 

14. The principle of virtual displacements is ad- 
mirably adapted to the acquisition of clearness and 
comprehensiveness in cases of this character, and we 
shall accordingly make use of it. To begin with, how- 
ever, let the following be noted. If the weight q (Fig. 
71) descend from position 1 to position 2, and a weight 
of exactly the same size move at the same time from 







THE PRINCIPLES OF STATICS. 



i to 3, the work performed in this operation is qh x -f- ^i'jl 
qh = q {h x -f h 2 ), the same, that is, as if the weight marks - 
q passed directly from i to 3 and the weight at 2 re- 
mained in its original position. The observation is 
easily generalised. 



/ 





Of 



3 

F'g 7 



4 



i 



*' ' '"* M 



\dh 



' i'- 



Fig- 



Let us consider a heavy homogeneous rectangular 
parallelepipedon, with vertical edges of the length //, 
base A, and the specific gravity s (Fig. 72). Let this 
parallelepipedon (or, what is the same thing, its centre 
of gravity) descend a distance dh. The work done is 
then Ahs.dh, or, also, A dhs .h. In the first expres- 
sion we conceive the whole weight A As displaced the 
vertical distance dh ; in the second we conceive the 
weight A dhs as having descended from the upper 
shaded space to the lower shaded space the distance //, 
and leave out of account 
the rest of the body. 
Both methods of concep- 
tion are admissible and 
equivalent. 

15. With the aid of 
this observation we shall 
obtain a clear insight into 
the paradox of Pascal, which consists of the following. 
The vessel g (Fig. 73), fixed to a separate support and 
consisting of a narrow upper and a very broad lower 
cylinder, is closed at the bottom by a movable piston, 




102 THE SCIENCE OF MECHANICS. 

which, by means of a string passing through the axis 
of the cylinders, is independently suspended from the 
extremity of one arm of a balance. If g be filled with 
water, then, despite the smallness of the quantity of 
water used, there will have to be placed on the other 
scale-pan, to balance it, several weights of consider- 
able size, the sum of which will be A/is, where A is 
the piston-area, h the height of the liquid, and s its 
specific gravity. But if the liquid be frozen and the 
mass loosened from the walls of the vessel, a very 
small weight will be sufficient to preserve equilibrium. 
The expia- Let us look to the virtual displacements of the two 

nation of ._. _ . _ ... 

the paradox cases (rig. 74). In the first case, supposing the pis- 
ton to be lifted a distance dh, the virtual moment is 

Adhs.h ox Ahs.dh. It thus 

comes to the same thing, 

dh whether we consider the mass 

that the motion of the piston 

t^h displaces to be lifted to the 



dh 






"M>>"\ 



upper surface of the fluid 



Flg - 74, through the entire pressure- 

height, or consider the entire weight A /is lifted the 
distance of the piston-displacement dh. In the second 
case, the mass that the piston displaces is not lifted to 
the upper surface of the fluid, but suffers a displace- 
ment which is much smaller the displacement, namely, 
of the piston. If A, a are the sectional areas respect- 
ively of the greater and the less cylinder, and k and / 
their respective heights, then the virtual moment of the 
present case is Ad /is. k + adhs./=(Ah-{- a I) s. dh; 
which is equivalent to the lifting of a much smaller 
weight (Ah -)- a/) s, the distance dh. 

16. The laws relating to the lateral pressure of 
liquids are but slight modifications of the laws of basal 



THE PRIXCIPLES OF STATICS. 



103 




pressure. If we have, for example, a cubical vessel The laws of 
of 1 decimetre on the side, which is a vessel of litre pressure, 
capacity, the pressure on any one of the vertical lateral 
walls ABCD, when the vessel is filled with water, is 
easily determinable. The deeper the migratory element 
considered descends beneath the surface, the greater 
the pressure will be to which it is subjected. We easily 
perceive, thus, that the pressure on a lateral wall is rep- 
resented by a wedge of water ABCDHI resting upon 
the wall horizontally 
placed, where ID is at 
right angles to BD and 
ID = HC=AC. The 
lateral pressure accor- 
dingly is equal to half 
a kilogramme. 

To determine the 
point of application of the resultant pressure, conceive 
ABCD again horizontal with the water-wedge resting 
upon it. We cut off AK= BL =\AC, draw the 
straight line KL and bisect it at M '; M is the point of 
application sought, for through this point the vertical 
line cutting the centre of gravity of the wedge passes. 

A plane inclined figure forming the base of a vessel The pres- 
filled with a liquid, is divided into the elements a, a', plane in- 
a" . . . with the depths h, ft ', h" . . . below the level of c 
the liquid. The pressure on the base is 

{ah + a' ft -f a" ft' -j- . . .) s. 
If we call the total base-area A, and the depth of its 
centre of gravity below the surface H, then 

ah -f- a'h' -f a"h" + . . . ah -f a'h' + . . 



Fig 75- 



a -j- a' -j- a" + . . . A 

whence the pressure on the base is AHs. 



1 =H, 





::- 

: " I -: \ : - - ' : ----- z\~- \- - - : r : . : - : . - 
ir : .5 "}.-_ - A;:rr the i^-zt: :: 5:^ ~.i .-: .5 
I:-.:- : - :z ti-_e r_:r: :r :: the . : : 1 ;:".:- ::" it 
?:. i~zi . 5 t : rt :~ ~:~ i- ; z .:- " ... :- ~zz 
ported by the rirrumnatant liquid.* The resultant of 
the ::::._- ::" r/ressure i::.:: : - :- r_rth:es - i:::r- 
iizzz'.y applied a: the centre ::" z"-.:zy ol the liquid 
placed by the solidified body, and is equal and opposite 
:: :- - T- T h: I: - : e : -t - the 7.1:^ :: the s-: :- 

ifced liquid another different body of the same form, bat 
: : 1 ; ~ : . ~ t ~t -t :.~ : ~r.v : z:.t : ; : : t^ :: tress _re it 
: : :::::-...::-.::: the fi~t A::::;^;/.. :^:t 
-:--=;: :- :Ar : : : : : thrtes. the "e z'z.z :: ti e : : : . 
zzz. - : it the ter tre :: _ri- :; :: :'. :e : : ir_ 1 the ;p- 
ir: : -: i- :;-. the re^c tit :: ti - - ;ti ze-t ress .res. 
;:::e:a:::.: ter. tre :.ri : zz tite :.-:.!::: iitnid. 
. ".: : : -~~-~ :: zzz' : ;rz : est :~ .: zzz.ie :n. ;~ 
: e zise : h :-:::t-e : is s : hi iz : -iies 

- 
titade A and base a, with edges vertically placed, 
::. : :: = tze:itt: _ri :; then the zr-.~-.z-: :z 
upper Z2S<*1 surface, when at a depth k below the] 
off the liquid is mks, while the ptessme on the 
- _r: \ :- :s . :. As the .iztzz. ::--: .re~ 
each other, an excess of pressure a is upwards 
mains: or, where r denotrs the l o lnm e of the 
ieler :tzeizn. an excess r, k 

We shall app ro ach n carrst die ffwdifli 
t:: : n z: . v. t.t Arc--;rr.e tes stirte 1. . ::. 

: he zzrir: tithe :: rtuii i stziiterrerts Let : 
leheritredzr. Fin. ~z :: the srez:h: zzz v ' ' z --- 
and height I sink the distance dk. The virtual 
merit : the trir.szerer: ze :r :~ the .::-: :r.: he i 
shiieh stite zf the urure ~ : ce _- -' ': f : 5 _t "a 






THE PRIXCIPLES OF STATICS. 



105 



his is done, the liquid rises from the lower into the up- 
>er space, and its moment is adhsh. The total vir- 
ual moment is therefore ah (ff s) dh = (/> q) dh, 
fhere/ denotes the weight of the body and q the weight 
if the displaced liquid. 

B 





Fig. 76. Fig. 77. 

18. The question might occur to us, whether the 
pward pressure of a body in a liquid is affected by the 
mmersion of the latter in another liquid. As a fact, 
his very question has been proposed. Let therefore 
Fig. 77) a body K be submerged in a liquid A and the 
iquid with the containing vessel in turn submerged in 
nother liquid B. If in the determination of the loss 
if weight in A it were proper to take account of the 
oss of weight of A in B, then K's loss of weight would 
lecessarily vanish when the fluid B became identical 
yith A. Therefore, A' immersed in A would suffer a 
oss of weight and it would suffer none. Such a rule 
vould be nonsensical. 

With the aid of the principle of virtual displace- 
ments, we easily comprehend the more complicated 
ases of this character. If a body be first gradually 
mmersed in B, then partly in B and partly in A, 
inally in A wholly ; then, in the second case, consider- 
ng the virtual moments, both liquids are to be taken 
nto account in the proportion of the volume of the 
)ody immersed in them. But as soon as the body is 
vholly immersed in A, the level of A on further dis- 



Is the buoy- 
ancy of a 
body in a 
liquid af- 
fected by 
the immer- 
sion ot that 
liquid in a 
second 
liquid? 



The eluci- 
dation of 
more com- 
plicated 
casesof this 
class. 



io6 



THE SCIENCE CF MECHANICS. 



The coun- 
ter-experi- 
ment. 



placement no longer rises, and therefore B is no longer 
of consequence. 
TheArchi- 19. Archimedes's principle may be illustrated by a 

medean . . 

principle ii- pretty experiment. From the one extremity of a scale- 

lustratedby . 

anexperi- beam (Fig. 78) we hang a hollow cube H. and beneath 

msnt. ' m 

it a solid cube M, which exactly fits into 
the first cube. We put weights into the 
opposite pan, until the scales are in 
equilibrium. If now M be submerged 
in water by lifting a vessel which stands 
beneath it, the equilibrium will be dis- 
turbed ; but it will be immediately re- 
stored if H, the hollow cube, be filled 
with water. 

A counter-experiment is the follow- 
ing. H is left suspended alone at the 
one extremity of the balance, and into 
the opposite pan is placed a vessel of 
water, above which on an independent 
support J/hangs by a thin wire. The scales are brought 
to equilibrium. If now M be lowered until it is im- 
mersed in the water, the equilibrium of the scales will 
be disturbed ; but on filling H with water, it will be 
restored. 

At first glance this experiment appears a little para- 
doxical. We feel, however, instinctively, that M can- 
not be immersed in the water without exerting a pres- 
sure that affects the scales. When we reflect, that the 
level of the water in the vessel rises, and that the solid 
body M equilibrates the surface-pressure of the water 
surrounding it, that is to say represents and takes the 
place of an equal volume of water, it will be found 
that the paradoxical character of the experiment van- 
ishes. 




Fig. 78. 



Remarks on 
the experi- 
ment. 



THE PRIXCIPLES OF STATICS. 107 

20. The most important statical principles have The gene- 
been reached in the investigation of solid bodies. This pies of stat- 

1 1 1 1 j 1 - 1 ' cs ight 

course is accidentally the historical one, but it is by no have been 
means the only possible and necessary one. The dif- the investi- 
ferent methods that Archimedes, Stevinus, Galileo, and fluid bodies 
the rest, pursued, place this idea clearly enough before 
the mind. As a matter of fact, general statical princi- 
ples, might, with the assistance of some very simple 
propositions from the statics of rigid bodies, have been 
reached in the investigation of liquids. Stevinus cer- 
tainly came very near such a discover} - . We shall stop 
a moment to discuss the question. 

Let us imagine a liquid, the weight of which we neg- The dis- 
lect. Let this liquid be enclosed in a vessel and sub- illustration 
jected to a definite pressure. A portion of the liquid, statement, 
let us suppose, solidifies. On the closed surface nor- 
mal forces act proportional to the elements of the area, 
and we see without difficulty that their resultant will 
always be = 0. 

If we mark off by a closed curve a portion of the 
closed surface, we obtain, on either side of it, a non- 
closed surface. All surfaces which are bounded by the 
same curve (of double curvature) and on which forces 
act normally (in the same sense) pro- \ W / 

portional to the elements of the area, \^ *x^ 
have lines coincident in position for 7 1* 

the resultants of these forces. >X \f 

Let us suppose, now, that a fluid /V X 

cylinder, determined by any closed '/"T\ 

plane curve as the perimeter of its Fig. 79. 

base, solidifies. We may neglect the two basal sur- 
faces, perpendicular to the axis. And instead of the 
cylindrical surface the closed curve simply may be con- 
sidered. From this method follow quite analogous 




108 THE SC/EXCE OF MECHAXICS, 

The dis- propositions for normal forces proportional to the ele- 

cassioo and . . 

illustration ments of a plane curve. 

statement. If the closed curve pass into a triangle, the con- 
sideration will shape itself thus. The resultant normal 
forces applied at the middle points of the sides of the 
triangle, we represent in direction, sense, and magni- 
tude by straight lines (Fig. 80). The 
lines mentioned intersect at a point 
the centre of the circle described about 
the triangle. It will further be noted, 
Fig. 80. thaf by t jj e s i m pi e parallel displace- 

ment of the lines representing the forces a triangle is 
constructible which is similar and congruent to the 
original triangle. 
Thedednc- Thence follows this proposition : 
triangle of Any three forces, which, acting at a point, are pro- 
this method port ional and parallel in direction to the sides of a tri- 
angle, and which on meeting by parallel displacement 
form a congruent triangle, are in equilibrium. We see 
at once that this proposition is simply a different form 
of the principle of the parallelogram of forces. 

If instead of a triangle we imagine a polygon, we 
shall arrive at the familiar proposition of the polygon 
of forces. 

We conceive now in a heavy liquid of specific gravity 
x a portion solidified. On the element a of the closed 
encompassing surface there acts a normal force a x x f 
where s is the distance of the element from the level of 
the liquid. We know from the outset the result. 
similar de- If normal forces which are determined by axs, 
another im- where a denotes an element of area and s its perpen- 
position. dicular distance from a given plane E, act on a closed 
surface inwards, the resultant will be V. x, in which ex- 
pression V represents the enclosed volume. The 




THE PRiyCIPLJ nog 

5 at the centre of gravity 
.ar to the plane mentioned 
irds this plane. 

ier the same conditions let a rigid curved surface The propo- 

:a here 

be bounded by a plane curve, which encloses on the 

plane the area A. The resultant of the forces acting agGrms 

on the curved surface is R r where 

M* = . - cos v. 

in which expression Z denotes the distance of the 
centre of gravity of the surface A from E r and v the 
a] angle of E and A. 
In the proposition of the last paragraph mathe- 
matically practised readers will have recognised a par- 
r case of Green's Theorem, whichi consists in the 
i of snrface-integratio ns to voiume-nfcegra- 

We may. accordingly, see into the force-system 
in eqpmEbrimm, or, if you plea- 
tems of forces of greater or less compl 
reach by a short path steriori. It is a 

mere accident that Stev; not light on tht 

propositions. The method here pursued correspc 
exactly to his. In this manner new discoveries can 
still be made. 

The paradoxical results that were reached in Fruitful re- 

* . ..... - t suits of the 

the investigation of liquids, supplied a stimulus to tur- invest . 
ther lefiection and research. It should also not be left domain, 
unnoticed, that the conception of a pkysico-mechanxad 
continuum was first formed on the occasion of the in- 
vestigation of liquids. A much freer and much more 
fruitful mathematical mode of view was developed 
thereby, than was possible through the study even : : 



no THE SCIENCE OF MECHANICS, 

systems of several solid bodies. The origin, in fact, 
of important modern mechanical ideas, as for instance 
that of the potential, is traceable to this source. 



VII. 

THE PRINCIPLES OF STATICS IN THEIR APPLICATION TO 
GASEOUS BODIES. 

character i. The same views that subserve the ends of science 

of this de- . ... . 

panmentof m the investigation of liquids are applicable with but 

inquiry. 

slight modifications to the investigation of gaseous 
bodies. To this extent, therefore, the investigation of 
gases does not afford mechanics any very rich returns. 
Nevertheless, the first steps that were taken in this 
province possess considerable significance from the 
point of view of the progress of civilisation and so 
have a high import for science generally. 
The eius- Although the ordinary man has abundant oppor- 

its subject- tunity, by his experience of the resistance of the air, by 
the action of the wind, and the confinement of air in 
bladders, to perceive that air is of the nature of a body, 
yet this fact manifests itself infrequently, and never in 
the obvious and unmistakable way that it does in the 
case of solid bodies and fluids. It is known, to be sure, . 
but is not sufficiently familiar to be prominent in popu-j 
lar thought. In ordinary life the presence of the air is 
scarcely ever thought of. (See p. 517.) 
The effect Although the ancients, as we may learn from thf 

of the first . TT . . . . 1 i 

disclosures accounts of Vitruvius, possessed instruments which 
Ince. 1 P v like the so-called hydraulic organs, were based on th< 
condensation of air, although the invention of the air 
gun is traced back to Ctesibius, and this instrumen 
was also known to Guericke, the notions which peopl 
held with regard to the nature of the air as late eve 



THE PRIXCIPLES OF STATICS. 




OTTO dc GUERICKE 

Serenifs.^ Potential Elector i Brandeb 
Confiliiu-ius Civitat Ata^deb.Conlul 



H2 THE SCIENCE OF MECHANICS. 

as the seventeenth century were exceedingly curious 
and loose. We must not be surprised, therefore, at the 
intellectual commotion which the first more important 
experiments in this direction evoked. The enthusiastic 
description which Pascal gives of Boyle's air-pump ex- 
periments is readily comprehended, if we transport our- 
selves back into the epoch of these discoveries. What 
indeed could be more wonderful than the sudden dis- 
covery that a thing which we do not see, hardly feel, 
and take scarcely any notice of, constantly envelopes 
us on all sides, penetrates all things ; that it is the most 
important condition of life, of combustion, and of gi- 
gantic mechanical phenomena. It was on this occa- 
sion, perhaps, first made manifest by a great and strik- 
ing disclosure, that physical science is not restricted 
to the investigation of palpable and grossly sensible 
processes. AdcPV - $*ff. )/ 
The views 2. In Galileo's time philosophers explained the 

entertained , , . . , , , . , , 

on this sub- phenomenon of suction, the action of syringes and : 
leo's'time. 1 pumps by the so-called horror vacui nature's abhor- 
rence of a vacuum. Nature was thought to possess j 
the power of preventing the formation of a vacuum by i 
laying hold of the first adjacent thing, whatsoever it 
was, and immediately filling up with it any empty space 
that arose. Apart from the ungrounded speculative ! 
element which this view contains, it must be conceded, ' 
that to a certain extent it really represents the phe- ; 
nomenon. The person competent to enunciate it must ! 
actually have discerned some principle in the phenom-. 
enon. This principle, however, does not fit all cases.; 
Galileo is said to have been greatly surprised at hearing 
of a newly constructed pump accidentally supplied 
with a very long suction-pipe which was not able t< 
raise water to a height of more than eighteen Italia: 



THE PRIXCIPLES OF STATICS. 113 

ells. His first thought was that the horror vacui (or the 
resistenza del vacuo) possessed a measurable power. The 
greatest height to which water could be raised by suc- 
tion he called altezza limitatissima. He sought, more- 
over, to determine directly the weight able to draw out 
of a closed pump-barrel a tightly fitting piston resting 
on the bottom. 

3. Torricelli hit upon the idea of measuring the Ton-iceiH's 

. experiment 

resistance to a vacuum by a column of mercury instead 
of a column of water, and he expected to obtain a col- 
umn of about J T of the length of the water column. 
His expectation was confirmed by the experiment per- 
formed in 1643 by Yiviani in the well-known manner, 
and which bears to-day the name of the Torricellian 
experiment. A glass tube somewhat over a metre in 
length, sealed at one end and filled with mercury, is 
stopped at the open end with the finger, inverted in a 
dish of mercury, and placed in a vertical position. Re- 
moving the finger, the column of mercury falls and re- 
mains stationary at a height of about 76 cm. By this 
experiment it was rendered quite probable, that some 
very definite pressure forced the fluids into the vacuum. 
What pressure this was, Torricelli very soon divined. 

Galileo had endeavored, some time before this, to Galileo's 
determine the weight of the air, by first weighing a weigh air. 
glass bottle containing nothing but air and then again 
weighing the bottle after the air had been partly ex- 
pelled by heat. It was known, accordingly, that the 
air was heavy. But to the majority of men the horror 
vacui and the weight of the air were very distantly 
connected notions. It is possible that in Torricelli's 
case the two ideas came into sufficient proximity to 
lead him to the conviction that all phenomena ascribed 
to the horror vacui were explicable in a simple and 



ii 4 THE SCIENCE OF MECHANICS. 

Atmospher- logical manner by the pressure exerted by the weight 

ic pressure 

discovered of a fluid column a column of air. Torricelli discov- 

by Torri- 
celli. ered, therefore, the pressure of the atmosphere ; he also 

first observed by means of his column of mercury the 
variations of the pressure of the atmosphere. 

4. The news of Torricelli's experiment was circu- 
lated in France by Mersenne, and came to the knowl- 
edge of Pascal in the year 1644. The accounts of. the 
theory of the experiment were presumably so imper- 
fect that Pascal found it necessary to reflect indepen- 
dently thereon. {Pcsanteur de Pair. Paris, 1663.) 
Pascal's ex- He repeated the experiment with mercury and with 

perirnents. , * 

a tube of water, or rather of red wine, 40 feet in length. 
He soon convinced himself by inclining the tube that 
the space above the column of fluid was really empty; 
and he found himself obliged to defend this view against 
the violent attacks of his countrymen. Pascal pointed 
out an easy way of producing the vacuum which they 
regarded as impossible, by the use of a glass syringe, 
the nozzle of which was closed with the finger under 
water and the piston then drawn back without much 
difficulty. Pascal showed, in addition, that a curved 
siphon 40 feet high filled with water does not flow, but 
can be made to do so by a sufficient inclination to the 
perpendicular. The same experiment was made on a 
smaller scale with mercury. The same siphon flows 
or does not flowaccording as it is placed in an inclined 
or a vertical position. 

In a later performance, Pascal refers expressly to 
the fact of the weight of the atmosphere and to the 
pressure due to this weight. He shows, that minute j 
animals, like flies, are able, without injury to them- 
selves, to stand a high pressure in fluids, provided only 
the pressure is equal on all sides ; and he applies this 



THE PRIXCIPLES OF ST A TICS. 



"5 




Fig. 81. 



at once to the case of fishes and of animals that live inTheanai- 
the air. Pascal's chief merit, indeed, is to have estab- liquid and 
lished a complete analogy between the phenomena con- ic pressure, 
ditioned by liquid pressure (water-pressure) and those 
conditioned by atmospheric pressure. 

5. By a series of experiments Pascal shows that 
mercury in consequence of atmospheric pressure rises 
into a space containing no air in the same way that, 
in consequence of water-pressure, it rises into a space 
containing no water. If into a deep ves- 
sel filled with water (Fig. 81) a tube be 
sunk at the lower end of which a bag of 
mercury is tied, but so inserted that the 
upper end of the tube projects out of the 
water and thus contains only air, then 
the deeper the tube is sunk into the water 
the higher will the mercury, subjected 
to the constantly increasing pressure of the water, as- 
cend into the tube. The experiment can also be made, 
with a siphon-tube, or with a tube open at its lower end. 

Undoubtedly it was the attentive consideration of The height 

.of moun- 

this very phenomenon that led Pascal to the idea that tains deter- 
mined by 
the barometer-column must necessarilv stand lower atthebarom- 

eter. 

the summit of a mountain than at its base, and that 
it could accordingly be employed to determine the 
height of mountains. He communicated this idea to 
his brother-in-law, Perier, who forthwith successfully 
performed the experiment on the summit of the Puy 
de Dome. (Sept. 19, 1648.) 

Pascal referred the phenomena connected with ad- Adhesion 

, . plates. 

hesion-plates to the pressure of the atmosphere, and 
gave as an illustration of the principle involved the re- 
sistance experienced when a large hat lying flat on a 
table is suddenly lifted. The cleaving of wood to the 



n6 



THE SCIENCE OF MECHANICS. 



A siphon 
which acts 
by water- 
pressure. 



Pascal's 

modifica- 
tion of the 
Torricelli- 
an experi- 
ment. 




Fig. 82. 



bottom of a vessel of quicksilver is a phenomenon of 
the same kind. 

Pascal imitated the flow produced in a siphon by 
atmospheric pressure, by the use of water-pressure. 
The two open unequal arms a and 
b of a three-armed tube a b c (Fig. 
82) are dipped into the vessels of 
mercury e and d. If the whole 
arrangement then be immersed in 
a deep vessel of water, yet so that 
the long open branch shall always 
project above the upper surface, 
the mercury will gradually rise in 
the branches a and b, the columns 
finally unite, and a stream begin to flow from the vessel 
d to the vessel e through the siphon-tube open above 
to the air. 

j The Torricellian experiment was modi- 

fied by Pascal in a very ingenious manner. 
A tube of the form abed (Fig. 83), oi 
double the length of an ordinary barom 
eter-tube, is filled with mercury. The, 1 
openings a and b are closed with the fin 
gers and the tube placed in a dish 
mercury with the end a downwards. I 
now a be opened, the mercury in ed wi 
all fall into the expanded portion at e, an 
the mercury in ab will sink to the heigl 
of the ordinary barometer-column. A va< 

r~Y uum is produced at b which presses tl 
I finger closing the hole painfully inward, 
Fig. 83. If b also be opened the column in a b w 
sink completely, while the mercury in the expandi, 
portion e, being now exposed to the pressure of t 



K3 



THE PRIXCIPLES OF STATICS. 117 

atmosphere, will rise in c d to the height of the barom- 
eter-column. Without an air-pump it was hardly pos- 
sible to combine the experiment and the counter- 
experiment in a simpler and more ingenious manner 
than Pascal thus did. 

6. With regard to Pascal's mountain-experiment, Supple- 
mentary re- 
we shall add the following brief supplementary remarks, marks on 

r3SC31 S 

Let b n be the height of the barometer at the level of mountain- 
experiment 
the sea, and let it fall, say, at an elevation of m metres, 

to kl) Q , where k is a proper fraction. At a further eleva- 
tion of m metres, we must expect to obtain the barom- 
eter-height /- . k b , since we here pass through a stratum 
of air the density of which bears to that of the first the 
proportion of k : 1. If we pass upwards to the altitude 
/* = //. /// metres, the barometer-height corresponding 
thereto will be 

A Un A 10g/> A h>g 

/i= loP (10g ^ _IOg ^ ) ' 

The principle of the method is, we see, a very simple 
one ; its difficulty arises solely from the multifarious 
collateral conditions and corrections that have to be 
looked to. 

7. The most original and fruitful achievements in The experi- 

,.1 j r r\ r* ments o! 

the domain of aerostatics we owe to Otto von CjUE-ottoyon 
ricke. His experiments appear to have been suggested 
in the main by philosophical speculations. He pro- 
ceeded entirely in his own way ; for he first heard of 
the Torricellian experiment from Valerianus Magnus 
at the Imperial Diet of Ratisbon in 1654, where he dem- 
onstrated the experimental discoveries made by him 
about 1650. This statement is confirmed by his method 



n8 THE SCIENCE OF MECHANICS. 

of constructing a water-barometer which was entirely 
different from that of Torricelli. 
Thehistori- Guericke's book (Experimenta nova, ut vocantur, 

cal value of x ' 

Guericke's Magdeburgica. Amsterdam. 1672) makes us realise 

book. . 

the narrow views men took in his time. The fact that 
he was able gradually to abandon these views and to 
acquire broader ones by his individual endeavor speaks 
favorably for his intellectual powers. We perceive 
with astonishment how short a space of time separates 
us from the era of scientific barbarism, and can no lon- 
ger marvel that the barbarism of the social order still 
so oppresses us. 
its specula- In the introduction to this book and in various other 
ter. places, Guericke, in the midst of his experimental in- 

vestigations, speaks of the various objections to the 
Copernican system which had been drawn from the 
Bible, (objections which he seeks to invalidate,) and 
discusses such subjects as the locality of heaven, the 
locality of hell, and the day of judgment. Disquisi- 
tions on empty space occupy a considerable portion 
of the work. 
Guericke's Guericke regards the air as the exhalation or odor 1 
the air. of bodies, which we do not perceive because we have ' 
been accustomed to it from childhood. Air, to him, j 
is not an element. He knows that through the effects of 1 
heat and cold it changes its volume, and that it is j 
compressible in Hero's Ball, or Pila Heronis ; on the 
basis of his own experiments he gives its pressure at j 
20 ells of water, and expressly speaks of its weight, by 
which flames are forced upwards. 

8. To produce a vacuum, Guericke first employed 
a wooden cask filled with water. The pump of a fire- 
engine was fastened to its lower end. The water, i' 
was thought, in following the piston and the action o 



THE PR/XC/PLES OF STATICS. 



119 




Guericke's First Experiments. (Experim. Magdeb.) 



THE SCIENCE OF MECHANICS. 



His at- 
tempts to 
produce a 
vacuum. 



His final 
success. 



gravity, would fall and be pumped out. Guericke ex- 
pected that empty space would remain. The fastenings 
of the pump repeatedly proved to be too weak, since in 
consequence of the atmospheric pressure that weighed 
on the piston considerable force had to be applied to 
move it. On strengthening the fastenings three power- 
ful men finally accomplished the exhaustion. But, 
meantime the air poured in through the joints of the 
cask with a loud blast, and no vacuum was obtained. 
In a subsequent experiment the small cask from which 
the water was to be exhausted was immersed in a larger 
one, likewise filled with water. But in this case, too, the 
water gradually forced its way into the smaller cask. 

Wood having proved in this way to be an unsuit- 
able material for the purpose, and Guericke having re- 
marked in the last experiment indications of success, 
the philosopher now took a large hollow sphere of 
copper and ventured to exhaust the air directly. At 
the start the exhaustion was successfully and easily 
conducted. But after a few strokes of the piston, the 
pumping became so difficult that four stalwart men 
(yiri quadrati), putting forth their utmost efforts, could i 
hardly budge the piston. And when the exhaustion 
had gone still further, the sphere suddenly collapsed, I 
with a violent report. Finally by the aid of a copper! 
vessel of perfect spherical form, the production of the' 
vacuum was successfully accomplished. Guericke de- 
scribes the great force with which the air rushed in on 
the opening of the cock. 

9. After these experiments Guericke constructed 
an independent air-pump. A great glass globular re 
ceiver was mounted and closed by a large detachabl< 
tap in which was a stop-cock. Through this openin; 
the objects to be subjected to experiment were place 



THE PRIXCIPLES OF STATICS. 121 

in the receiver. To secure more perfect closure the Guericke's 

air-pump. 

receiver was made to stand, with its stop-cock under 
water, on a tripod, beneath which the pump proper was 





Guericke's Air-pump. {Experim. Magdeb.) 

placed. Subsequently, separate receivers, connected 
with the exhausted sphere, were also employed in the 
experiments. 



122 THE SCIENCE OF MECHANICS. 

The curious The phenomena which Guericke observed with this 

phenomena r i i t>i 

observed by apparatus are manifold and various. The noise which 

means of . , ...... r , 

the air- water in a vacuum makes on striking the sides of the 
glass receiver, the violent rush of air and water into 
exhausted vessels suddenly opened, the escape on ex- 
haustion of gases absorbed in liquids, the liberation of 
their fragrance, as Guericke expresses it, were imme- 
diately remarked. A lighted candle is extinguished 
on exhaustion, because, as Guericke conjectures, it 
derives its nourishment from the air. Combustion, as 
his striking remark is, is not an annihilation, but a 
transformation of the air. 

A bell does not ring in a vacuum. Birds die in it. 
Many fishes swell up, and finally burst. A grape is kept 
fresh in vacuo for over half a year. 

By connecting with an exhausted cylinder a long 
tube dipped in water, a water-barometer is constructed. 
The column raised is 19-20 ells high; and Von Guericke 
explained all the effects that had been ascribed to the 
horror vacui by the principle of atmospheric pressure. 

An important experiment consisted in the weighing 
of a receiver, first when filled with air and then when 
exhausted. The weight of the air was found to vary 
with the circumstances ; namely, with the temperature 
and the height of the barometer. According to Gue- 
ricke a definite ratio of weight between air and water 
does not exist. 

The experi- But the deepest impression on the contemporary 
mentsrelat- , . . 

ingtoat- world was made by the experiments relating to atmos- \ 

mospheric . . . i i r j r 

pressure, pheric pressure. An exhausted sphere formed of two 
hemispheres tightly adjusted to one another was rent ; 
asunder with a violent report only by the traction of 
sixteen horses. The same sphere was suspended from , 



THE PRINCIPLES OF STATICS. 123 

a beam, and a heavily laden scale-pan was attached to 
the lower half. 

The cylinder of a large pump is closed by a piston. 
To the piston a rope is tied which leads over a pulley 
and is divided into numerous branches on which a 
great number of men pull. The moment the cylinder is 
connected with an exhausted receiver, the men at the 
ropes are thrown to the ground. In a similar manner 
a huge weight is lifted. 

Guericke mentions the compressed-air gun as some- Guericke's 

air-gun. 

thing already known, and constructs independently an 
instrument that might appropriately be called a rari- 
fied-air gun. A bullet is driven by the external atmos- 
pheric pressure through a suddenly exhausted tube, 
forces aside at the end of the tube a leather valve which 
closes it, and then continues its flight with a consider- 
able velocity. 

Closed vessels carried to the summit of a mountain ^^JcJf wert 
and opened, blow out air; carried down again in the *grf" demt 
same manner, they suck in air. From these and other/rmtetf as & 
experiments Guericke^iiscovers that the air is elastic. ^5 H S"^ 

10. The investigations of Guericke were continued The investi- 

. Rations of 

by an Englishman, Robert Boyle.* The new expen- Robert 

J & r Bovle. 

ments which Boyle had to supply were few. He ob- 
serves the propagation of light in a vacuum and the 
action of a magnet through it ; lights tinder by means 
of a burning glass ; brings the barometer under the re- 
ceiver of the air-pump, and was the first to construct 
a balance-manometer ["the statical manometer"]. 
The ebullition of heated fluids and the freezing of water 
on exhaustion were first observed by him. 

Of the air pump experiments common at the present 
day may also be mentioned that with falling bodies, 

And published by him in 1660, before the work of Von Guericke. Trans. 



I2 4 



THE SCIENCE OF MECHANICS. 



The fall of 
bodies in a 
vacuum. 



Quantita- 
tive data. 



The discov- 
ery of other 
gaseous 
substances. 



which confirms in a simple manner the view of Galileo 
that when the resistance of the air has been eliminated 
light and heavy bodies both fall with the same velo- 
city. In an exhausted glass tube a leaden bullet and a 
piece of paper are placed. Putting the tube in a ver- 
tical position and quickly turning it about a horizontal 
axis through an angle of 180 , both bodies will be seen 
to arrive simultaneously at the bottom of the tube. 

Of the quantitative data we will mention the fol- 
lowing. The atmospheric pressure that supports a 
column of mercury of 76 cm. is easily calculated from 
the specific gravity 13-60 of mercury to be 1-0336 kg. 
to 1 sq.cm. The weight of 1000 cu.cm. of pure, dry 
air at C. and 760 mm. of pressure at Paris at an ele- 
vation of 6 metres will be found to be 1 -293 grams, 
and the corresponding specific gravity, referred to 
water, to be 0-001293. 

11. Guericke knew of only one kind of air. We 
may imagine therefore the excitement it created when 
in 1755 Black discovered carbonic acid gas (fixed air) 
and Cavendish in 1766 hydrogen (inflammable air), 
discoveries which were soon followed by other similar 

ones. The dissimilar 
physical properties of 
gases are very strik- 
ing. Faraday has il- 
lustrated their great 
inequality of weight 
by a beautiful lecture- 
experiment. If from 
a balance in equilib- 
rium, we suspend (Fig. 84) two beakers A, B, the one 
in an upright position and the other with its opening 
downwards, we may pour heavy carbonic acid gas from 





Fig. 84. 



THE PRIXCIPLES OF STATICS. 125 

above into the one and light hydrogen from beneath 
into the other. In both instances the balance turns in 
the direction of the arrow. To-day, as we know, the 
decanting of gases can be made directly visible by the 
optical method of Foucault and Toeppler. 

1 2. Soon after Torricelli's discovery, attempts were The mercu- 

rial air- 
made to employ practically the vacuum thus produced, pump. 

The so-called mercurial air-pumps were tried. But no 
such instrument was successful until the present cen- 
tury. The mercurial air-pumps now in common use 
are really barometers of which the extremities are sup- 
plied with large expansions and so connected that their 
difference of level may be easily varied. The mercury 
takes the place of the piston of the ordinary air-pump. 

13. The expansive force of the air, a property ob- Boyiesiaw. 
served by Guericke, was more accurately investigated 

by Boyle, and, later, by Mariotte. The law which 
both found is as follows. If Fbe called the volume of 
a given quantity of air and P its pressure on unit area 
of the containing vessel, then the product V. P is 
always = a constant quantity. If the volume of the 
enclosed air be reduced one-half, the air will exert 
double the pressure on unit of area ; if the volume of 
the enclosed quantity be doubled, the pressure will 
sink to one-half ; and so on. It is quite correct as a 
number of English writers have maintained in recent 
times that Boyle and not Mariotte is to be regarded 
as the discoverer of the law that usually goes by 
Mariotte's name. Not only is this true, but it must 
also be added that Boyle knew that the law did not 
hold exactly, whereas this fact appears to have escaped 
Mariotte. 

The method pursued by Mariotte in the ascertain- 
ment cf the law was very simple. He partially filled 



126 



THE SCIENCE OF MECHANICS. 



His appa- 
ratus. 



Mariotte's Torricellian tubes with mercury, measured the volume 

experi- . . . 

ments. of the air remaining, and then performed the Torricel- 
lian experiment. The new volume of 
air was thus obtained, and by subtract- 
ing the height of the column of mer- 
cury from the barometer-height, also 
the new pressure to which the same 
quantity of air was now subjected. 

To condense the air Mariotte em- 
ployed a siphon-tube with vertical 
arms. The smaller arm in which the 
air was contained was sealed at the 
upper end ; the longer, into which the 
mercury was poured, was open at the 
upper end. The volume of the air 
was read off on the graduated tube, 
and to the difference of level of the 
mercury in the two arms the barometer- 
height was added. At the present day 
both sets of experiments are performed 
in the simplest manner by fastening a 
cylindrical glass tube (Fig. 86) rr t 
closed at the top, to a vertical scale 
and connecting it by a caoutchouc 
tube kk with a second open glass tube 
r'r', which is movable up and down 
the scale. If the tubes be partly filled 
with mercury, any difference of level 
whatsoever of the two surfaces of mer- 
cury may be produced by displacing 
r' ;', and the corresponding variations of volume of the 
air enclosed in r r observed. 

It struck Mariotte on the occasion of his investiga- 
tions that any small quantity of air cut off completely 




Fig. 86. 



THE PRIXCIPLES OF STATICS. 127 

from the rest of the atmosphere and therefore notTheexpan 

. sive force of 

directly affected by the latter's weight, also supported isolated 

the barometer-column ; as where, to give an instance, the atmos- 
phere, 
the open arm of a barometer-tube is closed. The simple 

explanation of this phenomenon, which, of course, 

Mariotte immediately found, is this, that the air before 

enclosure must have been compressed to a point at 

which its tension balanced the gravitational pressure 

of the atmosphere ; that is to say, to a point at which 

it exerted an equivalent elastic pressure. 

We shall not enter here into the details of the ar- 
rangement and use of air-pumps, which are readily 
understood from the law of Boyle and Mariotte. 

14. It simply remains for us to remark, that the dis- 
coveries of aerostatics furnished so much that was new 
and wonderful that a valuable intellectual stimulus pro- 
ceeded from the science. 



CHAPTER II. 



THE DEVELOPMENT OF THE PRINCIPLES OF 
DYNAMICS. 



Dynamics 
wholly a 
modern 
science. 







GALILEOS ACHIEVEMENTS. 

i. We now pass to the discussion of the funda- 
mental principles of dynamics. This is entirely a mod- 
ern science. The mechanical speculations of the an- 
cients, particularly of the Greeks, related wholly to 
statics. (Dynamics was founded by Galileo} We shall 
readily recognise the correctness of this assertion if we 
but consider a moment a few propositions held by the 
Aristotelians of Galileo's time. To explain the descent 
of heavy bodies and the rising of light bodies, (in li- 
quids for instance,) it was assumed that every thing and 
object sought its place : the place of heavy bodies was j 
below, the place of light bodies was above. Motions i 
were divided into natural motions, as that of descent, j 
and violent motions, as, for example, that of a pro- j 
jectile. From some few superficial experiments and j 
observations, philosophers had concluded that heavy i 
bodies fall more quickly and lighter bodies more slowly, 
or, more precisely, that bodies of greater weight fall 
more quickly and those of less weight more slowly. It 
is sufficiently obvious from this that the dynamical 
knowledge of the ancients, particularly of the Greeks, 
was very insignificant/ and that it was left to moderr 1 



THE PRIXCIPLES OF DYXAMICS. 



129 



times to lay the true foundations of this department of 
inquiry."^ (See Appendix, VII., p. 520.) .' 



^rtt 




2.1 1 he treatise Discorsi e danostrazioni maiematiche, ht t 1 
in which Galileo communicated to the world the first 7 \' 



130 THE SCIENCE OE MECHANICS. 

Galileo's dynamical investigation of the laws of falling bodies, 

investiga- 
tion of the appeared in 1638. The modern spirit that Galileo dis- 

laws of fall- . 1 1 r 

ing bodies, covers is evidenced here, at the very outset, by the tact 
that he does not ask why heavy bodies fall, but pro- 
pounds the question, How do heavy bodies fall ? in 
agreement with what law do freely falling bodies move? 
The method he employs to ascertain this law is this. 
He makes certain assumptions. He does not, however, 
like Aristotle, rest there, but endeavors to ascertain by 
trial whether they are correct or not.~] 

His first, The first theory on which he lights is the following. 

erroneous T . , . , .,11 r t r 11 1 j 

theory. It seems in his eyes plausible that a freely falling body, 
inasmuch as it is plain that its velocity is constantly 
on the increase, so moves that its velocity is double 
after traversing double the distance, and triple after 
traversing triple the distance ; in short, that the veloci-i 
ties acquired in the descent increase proportionally 
to the distances descended through. Before he pro ( 
ceeds to test experimentally this hypothesis, he reason: 
on it logically, implicates himself, however, in so doing 
in a fallacy. He says, if a body has acquired a certai: 
velocity in the first distance descended through, doubl 
the velocity in double such distance descended througl 
and so on ; that is to say, if the velocity in the secon 
instance is double what it is in the first, then the doub 
distance will be traversed in the same time as the ong 
nal simple distance. If, accordingly, in the case 
the double distance we conceive the first half tra 
ersed, no time will, it would seem, fall to the accou 
of the second half. The motion of a falling body a 
pears, therefore, to take place instantaneously; whi 
not only contradicts the hypothesis but also ocular e 
dence. We shall revert to this peculiar fallacy ! 
Galileo's later on. 



THE PRINCIPLES OF DYNAMICS. 131 

3. After Galileo fancied he had discovered this as- His second, 

. correct, as- 

sumption to be untenable, he made a second one, ac- sumption. 

cording to which the velocity acquired is proportional 
to the time of the descent. That is, if a body fall once, 
and then fall again during twice as long an interval of 
time as it first fell, it will attain in the second instance 
double the velocity it acquired in the first. He found 
no self-contradiction in this theory, and he accordingly 
proceeded to investigate by experiment whether the 
assumption accorded with observed facts. It was dif- 
ficult to prove by any direct means that the velocity 
acquired was proportional to the time of descent. It 
was easier, however, to investigate by what law the 
distance increased with the time ; and he consequently 
deduced from his assumption the relation that obtained 
between the distance and the time, and tested this by 
experiment. The deduction p Discussion 

r ^\ D and eluci- 

is simple, distinct, and per- ^ 

fectly correct. He draws 
(Fig. 87) a straight line, and 
on it cuts off successive por- Ch 
tions that represent to him Fig- 87. 

the times elapsed. At the extremities of these por- 
tions he erects perpendiculars (ordinates), and these 
represent the velocities acquired. Any portion OGoi 
the line OA denotes, therefore, the time of descent 
elapsed, and the corresponding perpendicular GH the 
velocity acquired in such time. 

If, now, we fix our attention on the progress of the 
velocities, we shall observe with Galileo the following 
fact : namely, that at the instant C, at which one-half 
OC of the time of descent OA has elapsed, the velocity 
CD is also one-half of the final velocity AB. 

If now we examine two instants of time, E and G, 




1 32 THE SCIENCE OF MECHANICS. 

Uniformly equally distant in opposite directions from the instant 

nccclcriitcd 

motion. C, we shall observe that the velocity HG exceeds the 
mean velocity CD by the same amount that EF falls 
short of it. For every instant antecedent to C there 
exists a corresponding one equally distant from it sub- 
sequent to C. Whatever loss, therefore, as compared 
with uniform motion with half the final velocity, is suf- 
fered in the first half of the motion, such loss is made 
up in the second half. . The distance fallen through we 
may consequently regard as having been uniformly de- 
scribed with half the final velocity. If, accordingly, 
we make the final velocity v proportional to the time 
of descent /, we shall obtain v-=gt, where g denotes 
the final velocity acquired in unit of time the so-called 
acceleration. The space s descended through is there- 
fore given by the equation s = (gl/2) t or s =gt 2 2. 
Motion of this sort, in which, agreeably to the assump- 
tion, equal velocities constantly accrue in equal inter- 
vals of time, we call uniformly accelerated motion. 
Tabieofthe If we collect the times of descent, the final veloci- 
iocities,and ties, and the distances traversed, we shall obtain the 

distances of r ,, . . . 

descent. following table : 

/. V. S. 



1. ijr- 1 x 1 

2. 2g. 2X2 

3. 3-. 3 X 3 

4. 4g. 4X4 



"t y n f V / - 



tg. / X ' 



2 



THE PRINCIPLES OF DYNAMICS. 133 

4. The relation obtaining between / and s admits Experiraeu- 

. . tal verifica- 

of experimental proof; and this Galileo accomplished tion of the 

law. 

in the manner which we shall now describe. 

We must first remark that no part of the knowledge 
and ideas on this subject with which we are now so 
familiar, existed in Galileo's time, but that Galileo had 
to create these ideas and means for us. Accordingly, 
it was impossible for him to proceed as we should do 
to-day, and he was obliged, therefore, to pursue a dif- 
ferent method. He first sought to retard the motion 
of descent, that it might be more accurately observed. 
He made observations on balls, which he caused to 
roll down inclined planes (grooves); assuming that only 
the velocity of the motion would be lessened here, but 
that the form of the law of descent would remain un- 
modified. If, beginning from the upper extremity, theTbearti- 
distances 1, 4, 9, 16 ... be notched off on the groove, ployed, 
the respective times of descent will be representable, 
it was assumed, by the numbers 1, 2, 3, 4 . . . ; a result 
which was, be it added, confirmed. The observation of 
the times involved, Galileo accomplished in a very in- 
genious manner. There were no clocks of the modern 
kind in his day : such were first rendered possible by 
the dynamical knowledge of which Galileo laid the 
foundations. The mechanical clocks which were used 
were very inaccurate, and were available only for the 
measurement of great spaces of time. Moreover, it 
was chiefly water-clocks and sand-glasses that were in 
use in the form in which they had been handed down 
from the ancients. Galileo, now, constructed a very 
simple clock of this kind, which he especially adjusted 
to the measurement of small spaces of time ; a thing 
not customary in those days. It consisted of a vessel of 
water of very large transverse dimensions, having in 



^ 



134 THE SCIENCE OF MECHANICS. 

Galileo's the bottom a minute orifice which was closed with the 
finger. As soon as the ball began to roll down the in- 
clined plane Galileo removed his finger and allowed the 
water to flow out on a balance ; when the ball had ar- 
rived at the terminus of its path he closed the orifice. 
As the pressure-height of the fluid did not, owing to 
the great transverse dimensions of the vessel, percept- 
ibly change, the weights of the water discharged from 
the orifice were proportional to the times. It was in 
this way actually shown that the times increased simply, 
while the spaces fallen through increased quadratically. 
The inference from Galileo's assumption was thus con- 
firmed by experiment, and with it the assumption itself. 

Thereia- 5. To form some notion of the relation which sub- 

tion of mo- . . , 

tion on an sists between motion on an inclined plane and that 01 
plane to free descent, Galileo made the assumption, that a body 

that of free ...... , , i-i r i j i 

descent, which falls through the height of an inclined plane 
attains the same final velocity as a body which falls 
through its length. This is an assumption that will 
strike us as rather a bold one ; but in the manner in 
which it was enunciated and employed by Galileo, it is 
quite natural. We shall endeavor to explain the way by 
which he was led to it. He says : If a body fall freely 
downwards, its velocity increases proportionally to the 
time. When, then, the body has arrived at a point be- 
low, let us imagine its velocity reversed and directed 
upwards ; the body then, it is clear, will rise. We make 
the observation that its motion in this case is a reflection, 
so to speak, of its motion in the first case. As then its 
velocity increased proportionally to the time of descent, 
it will now, conversely, diminish in that proportion. 
When the body has continued to rise for as long a 
time as it descended, and has reached the height from 
which it originally fell, its velocity will be reduced to 



THE PRINCIPLES OF DYNAMICS. 135 

zero. We perceive, therefore, that a body will rise, justifica- 

. ...... tion of the 

in virtue of the velocity acquired in its descent, just as assumption 

. mat the 

high as it has fallen. If, accordingly, a body falling final veioc- 

. . 1-1 itiesofsuch 

down an inclined plane could acquire a velocity which motions are 

. . the same. 

would enable it, when placed on a differently inclined 
plane, to rise higher than the point from which it had 
fallen, we should be able to effect the elevation of 
bodies by gravity alone. There is contained, accord- 
ingly, in this assumption, that the velocity acquired by 
a body in descent depends solely on the vertical height 
fallen through and is independent of the inclination of 
the path, nothing more than the uncontradictory ap- 
prehension and recognition of the/act that heavy bodies 
do not possess the tendency to rise, but only the ten- 
dency to fall. If we should assume that a body fall- 
ing down the length of an inclined plane in some way 
or other attained a greater velocity than a bod)' that 
fell through its height, we should only have to let the 
body pass with the acquired velocity to another in- 
clined or vertical plane to make it rise to a greater ver- 
tical height than it had fallen from. And if the velo- 
city attained on the inclined plane were less, we should 
only have to reverse the process to obtain the same re- 
sult. In both instances a heavy body could, by an ap- 
propriate arrangement of inclined planes, be forced 
continually upwards solely by its own weight a state 
of things which wholly contradicts our instinctive 
knowledge of the nature of heavy bodies. (See p. 522.) 

6. Galileo, in this case, again, did not stop with 
the mere philosophical and logical discussion of his 
assumption, but tested it by comparison with expe- 
rience. 

He took a simple filar pendulum (Fig. 88) with a 
heavy ball attached. Lifting the pendulum, while 



J 3 6 



THE SCIENCE OF MECHANICS. 



Galileo's elongated its full length, to the level of a given altitude, 
tai verifica- and then letting it fall, it ascended to the same level 

tion of this , . ..' Tr .. , . 

assumption on the opposite side. If it does not do so exactly, 
Galileo said, the resistance of the air must be the cause 
of the deficit. This is inferrible from the fact that the 
deficiency is greater in the case of a cork ball than it is 




Effected by in the case of a heavy metal one. However, this neg- 
fmpeding lected, the body ascends to the same altitude on the 
ofVpendu- opposite side. Now it is permissible to regard the mo- 
ums nng. ^^ ^ ^ pendulum in the arc of a circle as a motion 
of descent along a series of inclined planes of different 
inclinations. This seen, we can, with Galileo, easily 
cause the body to rise on a different arc on a different 
series of inclined planes. This we accomplish by driv- 
ing in at one side of the thread, as it vertically hangs, 
a nail/or^-, which will prevent. any given portion of 
the thread from taking part in the second half of the 
motion. The moment the thread arrives at the line of 
equilibrium and strikes the nail, the ball, which has 
fallen through ba, will begin to ascend by a different 
series of inclined planes, and describe the arc am ox an. 
Now if the inclination of the planes had any influence 



THE PRINCIPLES OF DYNAMICS. 137 

on the velocity of descent, the body could not rise to 
the same horizontal level from which it had fallen. 
But it does. By driving the nail sufficiently low down, 
we may shorten the pendulum for half of an oscillation 
as much as we please ; the phenomenon, however, al- 
ways remains the same. If the nail h be driven so low 
down that the remainder of the string cannot reach to 
the plane E, the ball will turn completely over and 
wind the thread round the nail ; because when it has 
attained the greatest height it can reach it still has a 
residual velocity left. 

7. If we assume thus, that the same final velocity is The as- 

' ' sumption 

attained on an inclined plane whether the bodv fall leads to the 

r J , law of rela- 

through the height or the length of the plane, in which tive accei- 
assumption nothing more is contained than that a body sought, 
rises by virtue of the velocity it has acquired in falling 
just as high as it has fallen, we shall easily arrive, 
with Galileo, at the perception that the times of the de- 
scent along the height and the length of an inclined 
plane are in the simple proportion of the height and 
the length ; or, what is the same, that the accelerations 
are inversely proportional to the times of descent. 
The acceleration along the height will consequently 
bear to the acceleration along ^ 
the length the proportion of the 
length to the height. Let AB 
(Fig. 89) be the height and AC &- 
the length of the inclined plane. Fig. 89. 

Both will be descended through in uniformly accel- 
erated motion in the times / and / 1 with the final ve- 
locity v. Therefore, 

. -r, v , . v AB t 
AB=- J tandAC= Y t 1 ,- iU = T . 




138 



THE SCIENCE OF MECHANICS. 



sniff. 



If the accelerations along the height and the length be 
called respectively g and g lt we also have 

<?", t AB 
v = gt and v = g x t x , whence == = ~ = 

g t \ At 

In this way we are able to deduce from the accel- 
eration on an inclined plane the acceleration of free 
descent. 
a corollary From this proposition Galileo deduces several cor- 

of the pre- 
ceding law. ollaries, some of which have passed into our elementary 

text-books. The accelerations along the height and 
length are in the inverse proportion of the height and 
length. If now we cause one body to fall along the 
length of an inclined plane and simultaneously another 
to fall freely along its height, and ask what the dis- 
tances are that are traversed by the two in equal inter- 
vals of time, the solution of the problem will be readily 
found (Fig. 90) by simply letting fall from B a perpen- 
dicular on the length. The part AD, thus cut off, will 
be the distance traversed by the one body on the in- 
clined plane, while the second body is freely falling 
through the height of the plane. 

A 




Fig. 90. Fig. 91. 

Relative If we describe (Fig. 91) a circle on AB as diame- 

scription of ter, the circle will pass through D, because D is a 
anddiame- right angle. It will be seen thus, that we can imagine 
cies. ' any number of inclined planes, AE, AF, of any degree 
of inclination, passing through A, and that in every 



THE PRIXCIPLES OF DYXAMICS. 



139 



case the chords AG, AH drawn in this circle from the 
upper extremity of the diameter will be traversed in 
the same time by a falling body as the vertical diame- 
ter itself. Since, obviously, only the lengths and in- 
clinations are essential here, we may also draw the 
chords in question from the lower extremity of the 
diameter, and say generally : The vertical diameter 
of a circle is described by a falling particle in the same 
time that any chord through either extremity is so 
described. 

We shall present another corollary, which, in the The figures 
pretty form in which Galileo gave it, is usually no bodies faV 

... ... T17 ing in the 

longer incorporated in elementary expositions. We chords of 
imagine gutters radiating in a vertical plane from a 
common point A at a 
number of different 
degrees of inclination 
to the horizon (Fig. 
92). We place at their 
common extremity A 
a like number of heavy 
bodies and cause them 
to begin simultaneous- 
ly their motion of des- 
cent. The bodies will 
always form at any one 

instant of time a circle. After the lapse of a longer time 
they will be found in a circle of larger radius, and the 
radii increase proportionally to the squares of the 
times. If we imagine the gutters to radiate in a space 
instead of a plane, the falling bodies will always form 
a sphere, and the radii of the spheres will increase pro- 
portionally to the squares of the times. This will be 




140 



THE SCIENCE OF MECHANICS. 



perceived by imagining the figure revolved about the 
vertical A V. 
Character 8. We see thus, as deserves again to be briefly 

inquiries, noticed, that Galileo did not supply us with a theory 
of the falling of bodies, but investigated and estab- 
lished, wholly without preformed opinions, the actual 
facts of falling. 

Gradually adapting, on this occasion, his thoughts 
to the facts, and everywhere logically abiding by the 
ideas he had reached, he hit on a conception, which to 
himself, perhaps less than to his successors, appeared 
in tne light of a new law. In all his reasonings, Galileo 
followed, to the greatest advantage of science, a prin- 
ciple which might appropriately be called the principle 
Theprin- of continuity . Once we have reached a theory that ap- 

ciple of . ITU 

continuity, plies to a particular case, we proceed gradually to 
modify in thought the conditions of that case, as far 
as it is at all possible, and endeavor in so doing to 
adhere throughout as closely as we can to the concep- 
tion originally reached. There is no method of pro- 
cedure more surely calculated to lead to that compre- 
hension of all natural phenomena which is the simplest 
and also attainable with the least expenditure of men- 
tality and feeling. (Compare Appendix, IX., p. 523.) 
A particular instance will show more clearly than 
any general remarks what we mean. Galileo con- 




B 



7/ 

Fig. 93- 

siders (Fig. 93) a body which is falling down the in- 
clined plane AB, and which, being placed with the 



THE PRINCIPLES OF DYNAMICS. 141 

velocity thus acquired on a second plane BC, for ex- Galileo's 

* . 11 1 discovery 

ample, ascends this second plane. On all planes BC, of the so- 

called law 

BD, and so forth, it ascends to the horizontal plane of inertia, 
that passes through A. But, just as it falls on BD 
with less acceleration than it does on BC, so similarly 
it will ascend on BD with less retardation than it will 
on BC. The nearer the planes BC, BD, BE, BF ap- 
proach to the horizontal plane BH, the less will the 
retardation of the body on those planes be, and the 
longer and further will it move on them. On the hori- 
zontal plane BH the retardation vanishes entirely (that 
is, of course, neglecting friction and the resistance of 
the air), and the body will continue to move infinitely 
long and infinitely far with constant velocity. Thus ad- 
vancing to the limiting case of the problem presented, 
Galileo discovers the so-called law of inertia, according 
to which a body not under the influence of forces, i. e. \j( 
of special circumstances that change motion, will re- 
tain forever its velocity (and direction). We shall 
presently revert to this subject. 

q. The motion of falling that Galileo found actually The deduc- 

. . . . .... tion of the 

to exist, is, accordingly, a motion of which the velocity idea of uni- 

.... 11 j fonnlvac- 

mcreases proportionally to the time a so-called uni- ceierated 
formly accelerated motion. 

It would be an anachronism and utterly unhistorical 
to attempt, as is sometimes done, to derive the uniformly 
accelerated motion of falling bodies from the constant 
action of the force of gravity. " Gravity is a constant 
force ; consequently it generates in equal elements of 
time equal increments of velocity ; thus, the motion 
produced is uniformly accelerated." Any exposition 
such as this would be unhistorical, and would put the 
whole discovery in a false light, for the reason that the 
notion of force as we hold it to-day was first created 



i 4 2 THE SCIENCE OF MECHANICS. 

Forces and by Galileo. Before Galileo force was known solely as 

accelera- 
tions, pressure. Now, no one can know, who has not learned 

it from experience, that generally pressure produces 
motion, much less in what manner pressure passes into 
motion ; that not position, nor velocity, but accelera- 
tion, is determined by it. This cannot be philosophi- 
cally deduced from the conception, itself. Conjectures 
may be set up concerning it. But experience alone can 
definitively inform us with regard to it. 

10. It is not by any means self-evident, therefore, 
that the circumstances which determine motion, that 
is, forces, immediately produce accelerations. A glance 
at other departments of physics will at once make this 
clear. The differences of temperature of bodies also 
determine alterations. However, by differences of tem- 
perature not compensatory accelerations are deter- 
mined, but compensatory velocities. 
The fact That it is accelerations which are the immediate ef- 

that forces - . . ... . . 

determine fects of the circumstances that determine motion, that 
tions is an is, of the forces, is a fact which Galileo perceived in the 
taifact. natural phenomena. Others before him had also per- 
ceived many things. The assertion that everything seeks 
its place also involves a correct observation. The ob- 
servation, however, does not hold good in all cases, 
and it is not exhaustive. If we cast a stone into the 
air, for example, it no longer seeks its place ; since its 
place is below. But the acceleration towards the earth, 
the retardation of the upward motion, the fact that Ga- 
lileo perceived, is still present. His observation always 
remains correct ; it holds true more generally ; it em- 
braces in one mental effort much more. 

ii. We have already remarked that Galileo dis- 
covered the so-called law of inertia quite incidentally. 
A body on which, as we are wont to say, no force acts, 



THE PRIXCIPLES OF D YNAM1CS. 143 

preserves its direction and velocity unaltered. The History of 
fortunes of this law of inertia have been strange. It called law 

. , , . . . . of inertia. 

appears never to have played a prominent part in Gali- 
leo's thought. But Galileo's successors, particularly 
Huygens and Newton, formulated it as an independent 
law. Nay, some have even made of inertia a general 
property of matter. We shall readily perceive, how-* 
ever, that the law of inertia is not at all an indepen- 
dent law, but is contained implicitly in Galileo's per- 
ception that all circumstances determinative of motion, 
or forces, produce accelerations. 

In fact, if a force determine, not position, not velo- The law a 
city, but acceleration, change of velocity, it stands toference 
reason that where there is no force there will be no leo-s fund*. 

. . . mental ob- 

change of velocity. It is not necessary to enunciate sedation, 
this in independent form. The embarrassment of the 
neophyte, which also overcame the great investigators 
in the face of the great mass of new material presented, 
alone could have led them to conceive the same fact as 
two different facts and to formulate it twice. 

In any event, to represent inertia as self-evident, or Erroneous 

... , ... , . methods of 

to derive it from the general proposition that "the ef- deducing it, 
feet of a cause persists," is totally wrong. Only a 
mistaken straining after rigid logic can lead us so out 
of the way. Nothing is to be accomplished in the pres- 
ent domain with scholastic propositions like the one 
just cited. We may easily convince ourselves that the 
contrary proposition, " cessante causa cessat effectus," 
is as well supported by reason. If we call the acquired 
velocity "the effect," then the first proposition is cor- 
rect ; if we call the acceleration "effect," then the sec- 
ond proposition holds. 

12. We shall now examine Galileo's researches from 
another side. He began his investigations with the 



i 4 4 



THE SCIENCE OF MECHANICS. 



Notion of notions familiar to his time notions developed mainly 

velocity as . . 

it existed in in the practical arts. One notion of this kind was that 
time. of velocity, which is very readily obtained from the con- 

sideration of a uniform motion. If a body traverse in 
every second of time the same distance c, the distance 
traversed at the end of / seconds will be j = ct. The 
distance c traversed in a second of time we call the ve- 
locity, and obtain it from the examination of any por- 
tion of the distance and the corresponding time by the 
help of the equation c = s/t, that is, by dividing the 
number which is the measure of the distance traversed 
by the number which is the measure of the time elapsed. 
Now, Galileo could not complete his investigations 
without tacitly modifying and extending the traditional 
idea of velocity. Let us represent for distinctness sake 




Fig. 94- 

in i (Fig. 94) a uniform motion, in 2 a variable motion, 
by laying off as abscissae in the direction OA the elapsed 
times, and erecting as ordinates in the direction AB the 
distances traversed. Now, in 1, whatever increment 
of the distance we may divide by the corresponding in- 
crement of the time, in all cases we obtain for the ve- 
locity c the same value. But if we were thus to proceed 
in 2, we should obtain widely differing values, and 
therefore the word "velocity " as ordinarily understood, 
ceases in this case to be unequivocal. If, however, we 
consider the increase of the distance in a sufficiently 



THE PRINCIPLES OF D YNAMICS. 145 

small element of time, where the element of the curve Galileo's 
in 2 approaches to a straight line, we may regard the tion of this 

...... notion. 

increase as uniform. The velocity in this element of 
the motion we may then define as the quotient, A s/A t, 
of the element of the time into the corresponding ele- 
ment of the distance. Still more precisely, the velocity 
at any instant is defined as the limiting value which 
the ratio A s/A t assumes as the elements become in- 
finitely small a value designated by ds jdt. This new 
notion includes the old one as a particular case, and is, 
moreover, immediately applicable to uniform motion. 
Although the express formulation of this idea, as thus 
extended, did not take place till long after Galileo, we 
see none the less that he made use of it in his reason- 
ings. 

13. An entirely new notion to which Galileo was The notion 

.... r i 1 ' acce ' era - 

led is the idea of acceleration. In uniformly acceler- tion. 
ated motion the velocities increase with the time 
agreeably to the same law as in uniform motion the 
spaces increase with the times. If we call v the velo- 
city acquired in time /, then vgt. Here g denotes 
the increment of the velocity in unit of time or the ac- 
celeration, which we also obtain from the equation 
g = v 't. When the investigation of variably accel- 
erated motions was begun, this notion of accelera- 
tion had to experience an extension similar to that of 
the notion of velocity. If in 1 and 2 the times be again 
drawn as abscissae, but now the velocities as ordinates, 
we may go through anew the whole train of the pre- 
ceding reasoning and define the acceleration as dv/dt> 
where dv denotes an infinitely small increment of the 
velocity and dt the corresponding increment of the 
time. In the notation of the differential calculus we 



146 



THE SCIENCE OF MECHANICS. 



have for the acceleration of a rectilinear motion, <p =. 
dv/Jt = d*s/di*. 

The ideas here developed are susceptible, moreover, 
of graphic representation. If we lay off the times as 
abscissae and the distances as ordinates, we shall per- 
ceive, that the velocity at each instant is measured by 
the slope of the curve of the distance. If in a similar 
manner we put times and velocities together, we shall 
see that the acceleration of the instant is measured by 
the slope of the curve of the velocity. The course of 
the latter slope is, indeed, also capable of being traced 
in the curve of distances, as will be perceived from 
the following considerations. Let us imagine, in the. 



The curve 
of distance, 





Fig. 95- 

usual manner (Fig. 95), a uniform motion represented 
by a straight line OCD. Let us compare with this a 
motion OCE the velocity of which in the second half 
of the time is greater, and another motion OCF of 
which the velocity is in the same proportion smaller. 
In the first case, accordingly, we shall have to erect for, 
the time OB = 2 OA, an ordinate greater than BD = 
1 AC; in the second case, an ordinate less than BD. 
We see thus, without difficulty, that a curve ot dis- 
tance convex to the axis of the time-abscissae corre 
sponds to accelerated motion, and a curve concave 
thereto to retarded motion. If we imagine a lead-pen 
cil to perform a vertical motion of any kind and ii 



THE PRIXCIPLES OF DYNAMICS. 147 

front of it during its motion a piece of paper to be uni- 
formly drawn along from right to left and the pencil to 
thus execute the drawing in Fig. 96, we shall be able to 
read off from the drawing the peculiarities of the mo- 
tion. At a the velocity of the pencil was directed up- 
wards, at b it was greater, at c it was =0, at d it was 
directed downwards, at e it was again = 0. At a, b, 
d, e, the acceleration was directed upwards, at c down- 
wards ; at c and e it was greatest. 

14. The summary representation of what Galileo Tabular 

. . . present- 

discovered is best made by a table of times, acquired ment of Ga- 
lileo's dis- 

/. v. s. covery ' 



1 


g 


If 


2 


% 


4 


3 


*g 


9 i 


/ 


*g 


/2 & 
9 



velocities, and traversed distances. But the numbers The table 
follow so simple a law, one immediately recognisable, placed by 

..... 1-1 rules for its 

that there is nothing to prevent our replacing the construc- 
table by a rule for its construction. If we examine the 
relation that connects the first and second columns, we 
shall And that it is expressed by the equation v = gt, 
which, in its last analysis, is nothing but an abbrevi- 
ated direction for constructing the first two columns 
of the table. The relation connecting the first and third 
columns is given by the equation s = g t 2 /i. The con- 
nection of the second and third columns is represented 
by s = v*/2g. 



148 THE SC/EXCE OF MECHANICS. 

rhe rules. Of the three relations 

v = gt 

S =~2T 
v 2 

strictly, the first two only were employed by Galileo. 
Huygens was the first who evinced a higher apprecia- 
tion of the third, and laid, in thus doing, the founda- 
tions of important advances. 
a remark 1 5. We may add a remark in connection with 

on the rela- . . , . , , , , . 

tionofthe this table that is very valuable. It has been stated 
the times, previously that a body, by virtue of the velocity it has 
acquired in its fall, is able to rise again to its origi- 
nal height, in doing which its velocity diminishes in 
the same way (with respect to time and space) as it 
increased in falling. Now a freely falling body ac- 
quires in double time of descent double velocity, but 
falls in this double time through four times the simple 
distance. A body, therefore, to which we impart a ver- 
tically upward double velocity will ascend twice as j 
long a time, but four times as high as a body to which 
the simple velocity has been imparted. 

The dispute It was remarked, very soon after Galileo, that there 
of the Car- ... . . . . , , 

tesians and is inherent in the velocity of a body a something that I 

iansonthe corresponds to a force a something, that is, by which 

measure of ."> 1 

force. a force can be overcome, a certain "efficacy, as it has 
been aptly termed. The only point that was debated ! 
was, whether this efficacy was to be reckoned propor- 
tional to the velocity or to the square of the velocity. 
The Cartesians held the former, the Leibnyzians the 
latter. But it will be perceived that the question in- 
volves no dispute whatever. The body with the double 
velocity overcomes a given force through double the 



THE PRINCIPLES OF D YNAMICS. 149 

time, but through four times the distance. With re- 
spect to time, therefore, its efficacy is proportional to 
the velocity : with respect to distance, to the square of 
the velocity. D'Alembert drew attention to this mis- 
understanding, although in not very distinct terms. It 
is to be especially remarked, however, that Huygens's 
thoughts on this question were perfectly clear. 

16. The experimental procedure by which, at the The present 

r x . expenmen- 

present dav, the laws of falling bodies are verified, istaimeansof 

- _ verifying 

somewhat different from that of Galileo. Two methods the laws of 

,. falling bod- 
may be employed. Either the motion of falling, which ies. 

from its rapidity is difficult to observe directly, is so 
retarded, without altering the law, as to be easily ob- 
served ; or the motion of falling is not altered at all, 
but our means of observation are improved in deli- 
cacy. On the first principle Galileo's inclined 
gutter and Atwood's machine rest. Atwood's 
machine consists (Fig. 97) of an easily run- 
ning pulley, over which is thrown a thread, 
to whose extremities two equal weights P are 
attached. If upon one of the weights P we 
lay a third small weight p, a uniformly accel- f\P 
erated motion will be set up by the added Fi s- 97- 
weight, having the acceleration {p/iP-\- f) g a result 
that will be readily obtained when we shall have dis- 
cussed the notion of "mass." Now by means of a 
graduated vertical standard connected with the pulley 
it may easily be shown that in the times 1, 2, 3, 4 ... . 
the distances 1, 4, 9, 16. . . . are traversed. The final 
velocity corresponding to any given time of descent is 
investigated by catching the small additional weight,/, 
which is shaped so as to project beyond the outline of 
P, in a ring through which the falling body passes, 
after which the motion continues without acceleration- 



ft 

P+p 



i 5 o THE SCIENCE OF MECHANICS. 

The apt>a- The apparatus of Morin is based on a different prin- 
Morin, La- ciple. A body to which a writing pencil is attached 
pich, and describes on a vertical sheet of paper, which is drawn 
uniformly across it by a clock-work, a horizontal straight 
line. If the body fall while the paper is not in motion, 
it will describe a vertical straight line. If the two 
motions are combined, a parabola will be produced, 
of which the horizontal abscissae correspond to the 
elapsed times and the vertical ordinates to the dis- 
tances of descent described. For the abscissae i, 2, 
3, 4 .... we obtain the ordinates 1, 4, 9, 16 ... . By 
an unessential modification, Morin employed instead of 
a plane sheet of paper, a rapidly rotating cylindrical 
drum with vertical axis, by the side of which the body 
fell down a guiding wire. A different apparatus, based 
on the same principle, was invented, independently, by 
Laborde, Lippich, and Von Babo. A lampblacked 
sheet of glass (Fig. 980) falls freely, while a horizon- 
tally vibrating vertical rod, which in its first transit 
through the position of equilibrium starts the motion 
of descent, traces, by means of a quill, a curve on the 
lampblacked surface. Owing to the constancy of the : 
period of vibration of the rod combined with the in- 
creasing velocity of the descent, the undulations traced 
by the rod become longer and longer. Thus (Fig. 98 
bcz=^ab, cd$ab, de = yab, and so forth. Tht 
law of falling bodies is clearly exhibited by this, since 
ab -\- cb = \ab, ab -f- be -f- cd = gab, and so forth 
The law of the velocity is confirmed by the inclination 
of the tangents at the points a, b, c, d, and so forth. I 
the time of oscillation of the rod be known, the valu 
of g is determinable from an experiment of this kin 
with considerable exactness. 

Wheatstone employed for the measurement of m 



THE PRIXCIPLES OF DYNAMICS. 



151 



nute portions of time a rapidly operating clock-work The de- 
called a chronoscope, which is set in motion at the be- wheat- 
ginning of the time to be measured and stopped at the Hipp, 
termination of it. Hipp has advantageously modified 



\ 




Fig. 98a. 



this method by simply causing a light index-hand to 
be thrown by means of a clutch in and out of gear with 
a rapidly moving wheel-work regulated by a vibrating 
reed of steel tuned to a high note, and acting as an es- 



152 



THE SCIENCE OF MECHANICS. 



Galileo's 
minor in- 
vestiga- 
tions. 



capement. The throwing in and out of gear is effected 
by an electric current. Now if, as soon as the body be- 
gins to fall, the current be interrupted, that is the hand 
thrown into gear, and as soon as the body strikes the 
platform below the current is closed, that is the hand 
thrown out of gear, we can read by the distance the 
index-hand has travelled the time of descent. 

17. Among the further achievements of Galileo we 
have yet to mention his ideas concerning the motion 
of the pendulum, and his refutation of the view that 
bodies of greater weight fall faster than bodies of less 
weight. We shall revert to both of these points on an- 
other occasion. It may be stated here, however, that 
Galileo, on discovering the constancy of the period of 
pendulum-oscillations, at once applied the pendulum 
to pulse-measurements at the sick-bed, as well as pro- 
posed its use in astronomical observations and to a cer- 
tain extent employed it therein himself. 
The motion 1 8. Of still greater importance are his investiga- 

of projec- . . ..... . . , 

tiles. tions concerning the motion of projectiles. A free body, 

according to Galileo's view, constantly experiences a 
vertical acceleration g towards the earth. If at the 
beginning of its motion it is affected with a vertical 
- v velocity c, its velocity at the 
end of the time / will be v = 
c -\-gt. An initial velocity up- \ 
wards would have to be reck- 
oned negative here. The dis- 
\ tance described at the end of j 
time / is represented by the 
equation s = a -\- c t -|~ \gt 2 , where ct and \gt 2 are the 
portions of the traversed distance that correspond re- 
spectively to the uniform and the uniformly accelerated 
motion. The constant a is to be put = when we reckon 



X 



Fig. 99- 



THE PRIXdPLES OF DYXAMICS. 153 

the distance from the point that the body passes at time 
/ = 0. When Galileo had once reached his fundamental 
conception of dynamics, he easily recognised the case 
of horizontal projection as a combination of two inde- 
pendent motions, a horizontal uniform motion, and a 
vertical uniformly accelerated motion. He thus intro- 
duced into use the principle of the parallelogram of mo- 
tions. Even oblique projection no longer presented the 
slightest difficulty. 

If a bodv receives a horizontal velocity c, it de- The curve 

... , . of projec- 

scribes in the horizontal direction in time / the distance tion a par- 
abola. 
y = ct, while simultaneously it falls in a vertical direc- 
tion the distance x = gt 2 /2. Different motion-deter- 
minative circumstances exercise no mutual effect on one 
another, and the motions determined by them take 
place independently of each other. Galileo was led to 
this assumption by the attentive observation of the 
phenomena ; and the assumption proved itself true. 

For the curve which a body describes when the two 
motions in question are compounded, we find, by em- 
ploying the two equations above given, the expression 
y = \/{2c 2 /g) x. It is the parabola of Apollonius hav- 
ing its parameter equal to c 2 jg and its axis vertical, 
as Galileo knew. 

We readily perceive with Galileo, that oblique pro- oblique 

j projection 

jection involves nothing new. The velocity c imparted 
to a body at the angle a with the horizon is resolvable 
into the horizontal component c . cos or and the vertical 
component c . sin a. With the latter velocity the body 
ascends during the same interval of time / which it 
would take to acquire this velocity in falling vertically 
downwards. Therefore, c. sin a =gt. When it has 
reached its greatest height the vertical component of 
its initial velocity has vanished, and from the point S 




154 THE SCIENCE OF MECHANICS. 

onward (Fig. ioo) it continues its motion as a horizon- 
tal projection. If we examine any two epochs equally 
distant in time, before and after the transit through S, 

we shall see that the body at 
these two epochs is equally 
distant from the perpendicu- 
lar through 5 and situated the 
same distance below the hori- 
zontal line through S. The 
Fi 8 I0 - curve is therefore symmet- 

rical with respect to the vertical line through S. It 
is a parabola with vertical axis and the parameter 
(c cos or) 2 /g. 
The range To find the so-called range of projection, we have 
o^projec- s j m p2y to consider the horizontal motion during the 
time of the rising and falling of the body. For the ascent 
this time is, according to the equations above given, 
t =z c sin a/g, and the same for the descent. With the 
horizontal velocity <r.cosar, therefore, the distance is 
traversed 

cs'xna e 2 . c 2 . 

w c cos a . 1 = 1 sin a cos a = sin 1 a. 

era- cr 

The range of projection is greatest accordingly 
when a = 45 , and equally great for any two angles I 
a = 45 /f. 
The mutual ig. The recognition of the mutual independence of 

indepen- . ... 

dence of the forces, or motion-determinative circumstances oc- 

forces 

curring in nature, which was 
reached and found expression 




in the investigations relating tc| 
^ projection, is important. A body 
Fg- ii. ma y move (Fig. 101) in the di 

rection AB, while the space in which this motion oc 
curs is displaced in the direction AC. The body ther 



THE PRIXCIPLES OF DYNAMICS. 155 

goes from A to D. Now, this also happens if the two 
circumstances that simultaneously determine the mo- 
tions AB and AC, have no influence on one another. 
It is easy to see that we may compound by the paral- 
lelogram not only displacements that have taken place 
but also velocities and accelerations that simultane- 
ously take place. (See Appendix, X., p. 525.) 



THE ACHIEVEMENTS OF HUYGENS. 

i. The next in succession of the great mechanical in- Huygens's 
quirers is Huygens, who in every respect must be as an in- 
ranked as Galileo's peer. If, perhaps, his philosophical qmr 
endowments were less splendid than those of Galileo, 
this deficiency was compensated for by the superiority 
of his geometrical powers. Huygens not only continued 
the researches which Galileo had begun, but he also 
solved the first problems in the dynamics of several 
masses, whereas Galileo had throughout restricted him- 
self to the dynamics of a single body. 

The plenitude of Huygens's achievements is bestEnumera- 

. . i-i j- l ' on ' Huy- 

seen in his HorologiumOscillatorium,. which appeared mgens's 

t>j j r 1 achieve- 

1073. a he most important subjects there treated 01 torments, 
the first time, are : the theory of the centre of oscilla- 
tion, the invention and construction of the pendulum- 
clock, the invention of the escapement, the determina- 
tion of the acceleration of gravity, g, by pendulum- 
observations, a proposition regarding the employment 
of the length of the seconds pendulum as the unit of 
length, the theorems respecting centrifugal .force, the 
mechanical and geometrical properties of cycloids, the 
doctrine of evolutes, and the theory of the circle of 
curvature. 



156 



THE SCIENCE OF MECHANICS. 



2. With respect to the form of presentation of his 
work, it is to be remarked that Huygens shares with 




CHRISTIANUS HUGENIUS 

natus 14. Aprilis 162$ . 
denatus 8 Junii 1635 . 

Illllllltlllllllllllll! 

Galileo, in all its perfection, the latter's exalted and 
inimitable candor. He is frank without reserve in the 
presentment of the methods that led him to his dis- 




THE PRIXCIPLES OF D YXAMICS. 



157 



coveries, and thus always 
conducts his reader into the 
full comprehension of his 
performances. Nor had he 
cause to conceal these 
methods. If, a thousand 
years from now, it shall be 
found that he was a man, it 
will likewise be seen what 
manner of man he was. 
In our discussion of the 
achievements of Huygens, 
however, we shall have to 
proceed in a somewhat dif- 
ferent manner from that 
which we pursued in the 
case of Galileo. Galileo's 
views, in their classical sim- 
plicity, could be given in an 
almost unmodified form. 
With Huygens this is not 
possible. The latter deals 
with more complicated 
problems; his mathematical 
methods and notations be- 
come inadequate and cum- 
brous. For reasons of brev- 
ity, therefore, we shall re- 
produce all the conceptions 
of which we treat, in mod- 
ern form, retaining, how- 
ever, Huygens's essential 
and characteristic ideas. 




Huygens's Pendulum Clock. 



158 THE SCIENCE OF MECHANICS. 

Centrifugal 3. We begin with the investigations concerning 

and centri- 
petal force, centrifugal force. When once we have recognised with 

Galileo that force determines acceleration, we are im- 
pelled, unavoidably, to ascribe every change of velocity 
and consequently also every change in the direction of 
a motion (since the direction is determined by three 
velocity-components perpendicular to one another) to 
a force. If, therefore, any body attached to a string, 
say a stone, is swung uniformly round in a circle, the 
curvilinear motion which it performs is intelligible only 
on the supposition of a constant force that deflects the 
body from the rectilinear path. The tension of the 
string is this force ; by it the body is constantly deflected 
from the rectilinear path and made to move towards 
the centre of the circle. This tension, accordingly, rep- 
resents a centripetal force. On the other hand, the axis 
also, or the fixed centre, is acted on by the tension of 
the string, and in this aspect the tension of the string 
appears as a centrifugal force. 





Fig. 102. Fig. 103. 

Let us suppose that we have a body to which a ve- 
locity has been imparted and which is maintained ir 
uniform motion in a circle by an acceleration constantly 
directed towards the centre. The conditions on whicl 
this acceleration depends, it is our purpose to investi 
gate. We imagine (Fig. 102) two equal circles un 



THE PRINCIPLES OF DYNAMICS. 159 

formly travelled round by two bodies ; the velocities in Uniform 

i-.xiTTi iii motion in 

the circles I and II bear to each other the proportion equal 

circles. 

1:2. If in the two circles we consider any same arc- 
element corresponding to some very small angle a, then 
the corresponding element s of the distance that the 
bodies in consequence of the centripetal acceleration 
have departed from the rectilinear path (the tangent), 
will also be the same. If we call q> x and q> the re- 
spective accelerations, and r and r/2 the time-elements 
for the angle a, we find by Galileo's law 

2s 2s 

9x= 2 '^2 = 4 2 , that is to say 9 2 =4^. 

Therefore, by generalisation, in equal circles the 
centripetal acceleration is proportional to the square of 
the velocity of the motion. 

Let us now consider the motion in the circles I and Uniform 

motion in 

II Fig. 103), the radii of which are to each other as unequal 

. circles. 

i : 2, and let us take for the ratio of the velocities of 
the motions also 1:2, so that like arc-elements are 
travelled through in equal times. <p x , <p 2 , s, 2s denote 
the accelerations and the elements of the distance trav- 
ersed ; t is the element of the time, equal for both 
cases. Then 

2s 4s 

<Pi ljfi><P% -,' that is to sa Y <Pi = 2( Pi- 

If now we reduce the velocity of the motion in II 
one-half, so that the velocities in I .and II become 
equal, <p 2 will thereby be reduced one-fourth, that is 
to say to <p x /2. Generalising, we get this rule: when 
the velocity of the circular motion is the same, the cen- 
tripetal acceleration is inversely proportional to the 
radius of the circle described. 

4. The early investigators, owing to their following 



i6o 



THE SCIEXCE OF MECHANICS. 



Deduction the conceptions of the ancients, generally obtained their 

of the gen- .... 

eraiiawof propositions in the cumbersome form of proportions. 

circular 

motion. We shall pursue a different method. On a movable 
object having the velocity v let a force act during the 
element of time r which imparts to the object perpen- 
dicularly to the direction of its motion the acceleration 
cp. The new velocity-component thus becomes q>r, 
and its composition with the first velocity produces a 
new direction of the motion, making the angle a with 
the original direction. From this results, by conceiving 
the motion to take place in a circle of radius r, and on 
account of the smallness of the angular element putting 





Fig. 104. Fig. 105. 

tan a a, the following, as the complete expression 

for the centripetal acceleration of a uniform motion in 

a circle, 

q>T vr v 2 

= tan a = a = or q> = . 
v r r 

The para- The idea of uniform motion in a circle conditioned 

doxical . ...... 

character by a constant centripetal acceleration is a little para- 
probiem. doxical. The paradox lies in the assumption of a con- 
stant acceleration towards the centre without actual 
approach thereto and without increase of velocity. This 
is lessened when we reflect that without this centripetal 
acceleration the body would be continually moving 
away from the centre ; that the direction of the accel- 



THE PRINCIPLES OF DYNAMICS. 161 

eration is constantly changing ; and that a change of 
velocity (as will appear in the discussion of the prin- 
ciple of vis viva) is connected with an approach of the 
bodies that accelerate each other, which does not take 
place here. The more complex case of elliptical cen- 
tral motion is elucidative in this direction. (See p. 527.) 

5. The expression for the centripetal or centrifugal a different 

. ' expression 

acceleration, q> = v 2 /r, can easily be put in a somewhat of the law. 
different form. If T denote the periodic time of the 
circular motion, the time occupied in describing the 
circumference, then vT= 2 r tt, and consequently cp = 
\m- 'T 1 , in which form we shall employ the expres- 
sion later on. If several bodies moving in circles have 
the same periodic times, the respective centripetal ac- 
celerations by which they are held in their paths, as is 
apparent from the last expression, are proportional to 
the radii. 

6. We shall take it for granted that the reader is Some phe- 

nomena 
familiar with the phenomena that illustrate the con- which the 

siderations here presented : as the rupture of strings of plains, 
insufficient strength on which bodies are whirled about, 
the flattening of soft rotating spheres, and so on. Huy- 
gens was able, by the aid of his conception, to explain 
at once whole series of phenomena. When a pendulum- 
clock, for example, which had been taken from Paris 
to Cayenne by Richer (1 671-1673), showed a retarda- 
tion of its motion, Huygens deduced the apparent 
diminution of the acceleration of gravity g thus estab- 
lished, from the greater centrifugal acceleration of the 
rotating earth at the equator ; an explanation that at 
once rendered the observation intelligible. 

An experiment instituted by Huygens may here be 
noticed, on account of its historical interest. When 
Newton brought out his theory of universal gravitation, 



R ' R 



162 THE SCIENCE OF MECHANICS. 

An interest- Huygens belonged to the great number of those who 

in experi- . 

ment of were unable to reconcile themselves to the idea of action 

Huygens. ... 

at a distance. He was of the opinion that gravitation 
could be explained by a vortical medium. If we enclose 
in a vessel filled with a liquid a number of lighter bod- 
ies, say wooden balls in water, and set the vessel ro- 
tating about its axis, the balls will at once rapidly move 
towards the axis. If for instance (Fig. 106), we place 
the glass cylinders RR containing the wooden balls KK 
by means of a pivot Z on a rotatory apparatus, and ro- 
tate the latter about its ver- 
"-*-> tical axis, the balls will im- 
mediately run up the cyl- | 
inders in the direction away 
nr from the axis. But if the 

A\ tubes be filled with water, 

Fi s- Io6 - each rotation will force the 

balls floating at the extremities EE towards the axis. 
The phenomenon is easily explicable by analogy wit! 
the principle of Archimedes. The wooden balls receivi 
a centripetal impulsion, comparable to buoyanc} 
which is equal and opposite to the centrifugal fore 
acting on the displaced liquid. (See p. 528.) 
Oscillatory 7. Before we proceed to Huygens's investigation 

motion. . 

on the centre of oscillation, we shall present to tl 
reader a few considerations concerning pendulous an' 
oscillatory motion generally, which will make up in o 
viousness for what they lack in rigor. 

Many of the properties of pendulum motion wi 
known to Galileo. That he had formed the conct - 
tion which we shall new give, or that at least he a 
on the verge of so doing, may be inferred from m< ) 
scattered allusions to the subject in his Dialogues. 1e 
bob of a simple pendulum of length / moves in a cii e 




THE PRIXCIPLES OF DYXAM/CS. 163 

(Fig. 107) of radius /. If we give the pendulum a very Galileo's 

..... , -ii investiga- 

small excursion, it will travel in its oscillations over ationofthe 
very small arc which coincides approximately with the pendulum- 
chord belonging to it. But this 
chord is described by a falling 
particle, moving on it as on an 
inclined plane (see Sect. 1 of this 
Chapter, 7), in the same time 
as the vertical diameter BD = 
2/. If the time of descent be 
called /, we shall have 2/ = 
hg* 2 , that is /=2V7/r. But 

. . , Fig. 107. 

since tne continued movement 

from B up the line BC occupies an equal interval of 
time, we have to put for the time T of an oscillation 
from Cto C, T= 41 / l/g. It will be seen that even from 
so crude a conception as this the correct form of the 
pendulum-laws is obtainable. The exact expression 
for the time of very small oscillations is, as we know, 

Again, the motion of a pendulum bob may be viewed pendulum 

. , f . 1 . , motion 

as a motion of descent on a succession ot inclined viewed as a 
planes. If the string of the pendulum makes the angle down in- 
a with the perpendicular, the pendulum bob receives planes, 
in the direction of the position of equilibrium the accel- 
eration g. sin a. When a is small, g. a is the expres- 
sion of this acceleration ; in other words, the accelera- 
tion is always proportional and oppositely directed to 
the excursion. When the excursions are small the 
curvature of the path may be neglected. 

8. From these preliminaries, we may proceed to 

the study of oscillatory motion in a simpler manner. A 

body is free to move on a straight line OA (Fig. 108), 

, and constantly receives in the direction towards the 



164 



THE SCIENCE OF MECHANICS. 



DC 




a simpler point an acceleration proportional to its distance from 

and modern 1 1 

view of os- 0. We will represent these accelerations by ordinates 

cillatory . . . 

motion. erected at the positions considered. Ordinates upwards 
denote accelerations towards the left ; ordinates down- 
wards represent accel- 
erations towards the 
right. The body, left 
to itself at A, will 
move towards O with 
varied acceleration, 
g'Jfi pass through O toA v 
where OA l = OA, 
come back to O, and 
so again continue its 
Fi - lo8 - motion. It is in the 

The period first place easily demonstrable that the period of os- 

ofoscilla- ..,..,. r . . . 

tion inde- cillation (the time of the motion through AOA t ) is m- 
theampii- dependent of the amplitude of the oscillation (the dis- 
tance OA). To show this, let us imagine in I and 
II the same oscillation performed, with single and 
double amplitudes of oscillation. As the acceleration 
varies from point to point, we must divide OA and \ 
O'A' = 20 A into a very large equal number of ele- 
ments. Each element A'B' of O'A' is then twice as 
large as the corresponding element AB of OA. The ; 
initial accelerations cp and cp stand in the relation 1 
cp' = 2 cp. Accordingly, the elements AB and A'B' = ; 
2AB are described with their respective accelerations; 
cp and 2(p in the same time r. The final velocities i\ 
and v in I and II, for the first element, will be #= (p \ 
and v' = 2<pr, that is v = 2 v. The accelerations anc 
the initial velocities at B and B' are therefore again a 
1 : 2. Accordingly, the corresponding elements tha 
next succeed will be described in the same time. An 



THE PRIXCIPLES OF D YNAMICS. 



165 



of every succeeding pair of elements the same asser- 
tion also holds true. Therefore, generalising, it will 
be readily perceived that the period of oscillation is 
independent of its amplitude or breadth. 

Next, let us conceive two oscillatory motions, I and The time of 

. oscilUtion 

II, that have equal excursions (Fig. 109); but in II let bmneij 

* v ' . proportion- 

a fourfold acceleration correspond to the same distance ai to the 

square root 

from O. We divide the amplitudes of 



I 



^3] 



\^^ 


BA 




/ 


II / 






/a 


BA' 



Fig. 109. 

v and v' at 



the oscillations AO and O 'A' = OA 
into a very large equal number of 
parts. These parts are then equal in 
I and II. The initial accelerations at 
A and A' are <p and 4 cp ; the ele- 
ments of the distance described are 
AB A'B' = s ; and the times are 
respectively rand r'. We obtain, then, 
r = 1 zs <p, r' = 1/2 j/4 <p = t/2. 
The element A'B' is accordingly trav- 
elled through in one-half the time 
the element AB is. The final velocities 
B and B' are found by the equations v = <pr and 
v' = 4 cp(r/2) = 2 v. Since, therefore, the initial velo- 
cities at B and B' are to one another as 1:2, and the 
accelerations are again as 1:4, the element of II suc- 
ceeding the first will again be traversed in half the 
time of the corresponding one in I. Generalising, we 
get : For equal excursions the time of oscillation is in- 
versely proportional to the square root of the accelera- 
tions. 

9. The considerations last presented may be put in 
a very much abbreviated and very obvious form by a 
method of conception first employed by Newton. New- 
ton calls those material systems similar that have geo- 
metrically similar configurations and whose homolo- 



of the ac- 
celeration. 



166 THE SCIENCE OF MECHANICS. 

The princi- gous masses bear to one another the same ratio. He 

pleofsimil- r ' , . . , 

nude. says further that systems ot this kind execute similar 
movements when the homologous points describe simi- 
lar paths in proportional times. Conformably to the 
geometrical terminology of the present day we should 
not be permitted to call mechanical structures of this 
kind (of five dimensions) similar unless their homolo- 
gous linear dimensions as well as the times and the 
masses bore to one another the same ratio. The struc- 
tures might more appropriately be termed affined to 
one another. 

We shall retain, however, the name phoronomically 
similar structures, and in the consideration that is to 
follow leave entirely out of account the masses. 
In two such similar motions, then, let 

the homologous paths be s and as, 

the homologous times be / and /3l; whence 

the homologous velo- 

s , a s 

cities are v = - and yv = y , 

the homologous accel- _ n 

Is Ot -S 

erations 9 = p and e( P = pt 71- 

Thededuc- Now all oscillations which a body performs under 
llwsof os* the conditions above set forth with any two different 
this a meUiod amplitudes i and a, will be readily recognised as sim- 
ilar motions. Noting that the ratio of the homologous 
accelerations in this case is = a, we have a = aj 'fi 2 . 
Wherefore the ratio of the homologous times, that is 
to say of the times of oscillation, is fi= i. We ob- 
tain thus the law, that the period of oscillation is inde- 
pendent of the amplitude. 

If in two oscillatory motions we put for the ratio 
between the amplitudes i : a, and for the ratio between 
the accelerations i : a /a, we shall obtain for this case 



THE PRIXCIPLES OF DYXAMICS. 167 



e= api= cclft 2 , and therefore /?== i/zfc y/ /i ; where- 
with the second law of oscillating motion is obtained. 

Two uniform circular motions are always phoronom- 
ically similar. Let the ratio of their radii be 1 : a and 
the ratio of their velocities 1 : y. The ratio of their 
accelerations is then = oc/fi 2 , and since y=a/fi, 
z\^oe=y 2 /a; whence the theorems relative to cen- 
tripetal acceleration are obtained. 

It is a pity that investigations of this kind respect- 
ing mechanical and phoronomical affinity are not more 
extensively cultivated, since they promise the most 
beautiful and most elucidative extensions of insight 
imaginable. 

10. Between uniform motion in a circle and oscil- The con- 
nection be- 
latory motion of the kind just discussed an important tweenoscii- 

1 m latory mo- 

relation exists which we shall now consider. We as- tion of this 

kind and 

sume a system of rectangular co- y uniform 

motion in a 

ordinates, having its origin at the / ^\ circle, 

centre, O, of the circle of Fig. no, 
about the circumference of which 
we conceive a body to move uni- 
formly. The centripetal accelera- 
tion q> which conditions this mo- 
tion, we resolve in the directions Fi s- IIG - 
of X and Y; and observe that the X-components of the 
motion are affected only by the X-components of the 
acceleration. We may regard both the motions and 
both the accelerations as independent of each other. 

Xow, the two components of the motion are os-Theiden- 

tityofthe 

cillatorv motions to and fro about O. To the excur- two. 
sion x the acceleration-component q> {xjr) or (jp/r) x 
in the direction O, corresponds. The acceleration is 
proportional, therefore, to the excursion. And accord- 
ingly the motion is of the kind just investigated. The 




168 THE SCIENCE OF MECHANICS. 

time T of a complete to and fro movement is also the 
periodic time of the circular motion. With respect to 
the latter, however, we know that (p= \r n 2 /T 2 , or, 
what is the same, that T= 2 7tV / r~/<p. Now cpjr is 
the acceleration for x= 1, the acceleration that corre- 
sponds to unit of excursion, which we shall briefly 
designate by f. For the oscillatory motion we may 
put, therefore, T= 2nyi/f. For a single movement 
to, or a single movement fro, the common method of 
reckoning the time of oscillation, we get, then, T= 
n 11//. 
Theappii- II. Now this result is directly applicable to pen- 

thehuure- dulum vibrations of very small excursions, where, ne- 
duiumvi- n "glecting the curvature of the path, it is possible to ad- 
rauons. ^ere to the conception developed. For the angle of 
elongation a we obtain as the distance of the pendulum 
bob from the position of equilibrium, la; and as the 
corresponding acceleration, ga; whence 

/=*"=* m &T=*Jl. 

lex I \ g 

This formula tells us, that the time of vibration is 
directly proportional to the square root of the length 
of the pendulum, and inversely proportional to the 
square root of the acceleration of gravity. A pendulum 
that is four times as long as the seconds pendulum, 
therefore, will perform its oscillation in two seconds. 
A seconds pendulum removed a distance equal to the 
earth's radius from the surface of the earth, and sub 
jected therefore to the acceleration g/ '4, will likewise 
perform its oscillation in two seconds. 

12. The dependence of the time of oscillation on I 
the length of the pendulum is very easily verifiable by 
experiment. If (Fig. in) the pendulums a, b, c, 



THE PRIXCIPLES OF DYXAMICS. 



169 



which to maintain the plane of oscillation invariable Experimen- 
are suspended by double threads, have the lengths 1, tfonofthe*" 
4, 9, then a will execute two oscillations to one oscil- pendulum 6 
lation of b, and three to one of c. 




Fig. III. 

The verification of the dependence of the time of 
oscillation on the acceleration of gravity g is some- 
what more difficult ; since the latter cannot be arbi- 
trarily altered. But the demonstration can be effected 
by allowing one component only of g to act on the 
pendulum. If we imagine the axis of oscillation of 



170 



THE SCIENCE OF MECHANICS. 



Experimen- the pendulum A A fixed in the vertically placed plane 

tal verinca- ..... 

tionofthe of the paper, EE will be the intersection of the plane 
pendulum. /ji of oscillation with the plane of the paper 
^^-y? and likewise the position of equilibrium 
of the pendulum. The axis makes with 
the horizontal plane, and the plane of os- 
cillation makes with the vertical plane, the 
angle fi; wherefore the acceleration g. cos ft 
is the acceleration which acts in this plane. 
If the pendulum receive in the plane of its oscillation 
the small elongation a, the corresponding acceleration 




Fig. 112. 




Fig. 113. 

will be Qj-cos/2) a; whence the time of oscillation is 
r= n Vljg cos/T 



THE PRIXCIPLES OF D YXAMICS. 



171 



We see from this result, that as ft is increased the 
acceleration ^cos/? diminishes, and consequently the 
time of oscillation increases. The experiment may be 
easily made with the apparatus represented in Fig. 113. 
The frame RR is free to turn about a hinge at C; it can 
be inclined and placed on its side. The angle of in- 
clination is fixed by a graduated arc G held by a set- 
screw. Every increase of ft increases the time of oscil- 
lation. If the plane of oscillation be made horizontal, 
in which position R rests on the foot F, the time of 
oscillation becomes infinitely great. The pendulum 
in this case no longer returns to any definite position 
but describes several complete revolutions in the same 
direction until its entire velocity has been destroyed 
by friction. 

13. If the movement of the pendulum do not take The conical 

. , _ , pendulum. 

place in a plane, but be performed in space, the thread 
of the pendulum will describe the surface 
of a cone. The motion of the conical pen- 
dulum was also investigated by Huygens. 
We shall examine a simple case of this 
motion. We imagine (Fig. 114) a pen- 
dulum of length / removed from the ver- 
tical by the angle a, a velocity v imparted 
to the bob of the pendulum at right 
angles to the plane of elongation, and the pendulum re- 
leased. The bob of the pendulum will move in a hori- 
zontal circle if the centrifugal acceleration <p developed 
exactly equilibrates the acceleration of gravity g; that 
is, if the resultant acceleration falls in the direction of 
the pendulum thread. But in that case cp/g= tan or. 
If T stands for the time taken to describe one revolu- 
tion, the periodic time, then cp = ^m^/T 2 or T = 
2 re yrjcp. Introducing, now, in the place of rjq> the 




Fig. 114. 



i 7 2 THE SCIENCE OF MECHANICS. 

value / sin a/g tan a = / cos a/g, we get for the periodic 
t time of the pendulum, T= i 7t V I cos a/g. For the ve- 
locity v of the revolution we find v = V rep, and since 
<p = gta.rx a it follows that v = V ' gl sin a tan a. For 
very small elongations of the conical pendulum we may 
put T=27r\///g, which coincides with the regular 
formula for the pendulum, when we reflect that a single 
revolution of the conical pendulum corresponds to two 
vibrations of the common pendulum. 
The deter- 1 4. Huygens was the first to undertake the exact 

tTe n acce l i- determination of the acceleration of gravity by means 
gravhyby of pendulum observations. From the formula 7'= 
ium!' en '" Tt V l/g for a simple pendulum with small bob we ob- 
tain directly g= n 2 l/T 2 . For latitude 45 we obtain 
as the value of g, in metres and seconds, 9 . 806. For 
provisional mental calculations it is sufficient to re- 
member that the acceleration of gravity amounts in 
round numbers to 10 metres a second. 
a remark 1 5. Every thinking beginner puts to himself the 

u"aex P reL question how it is that the duration of an oscillation, 
mgt e aw. t ^ t . g ^ f^ m g f can be found by dividing a number that 
is the measure of a length by a number that is the 
measure of an acceleration and extracting the square 
root of the quotient. But the fact is here to be borne in 
mind that g= 2 s/l 2 , that is a length divided by the 
square of a time. In reality therefore the formula we 
have is T n \/(l/2s)l 2 . And since Ijzs is the ratio 
of two lengths, and therefore a number, what we have 
under the radical sign is consequently the square of a 
time. It stands to reason that we shall find T in sec- 
onds only when, in determining g, we also take the sec- 
ond as unit of time. 

In the formula g= n 2 l/T 2 we see directly that g is 



of the prob- 



TIIE PRINCIPLES OF DYNAMICS. 173 

a length divided by the square of a time, according to 
the nature of an acceleration. 

16. The most important achievement of H uy gens The prob- 

. ... lemofthe 

is his solution of the problem to determine the centre centre of 

. oscillation. 

of oscillation. So long as we have to deal with the dy- 
namics of a single body, the Galilean principles amply 
suffice. But in the problem just mentioned we have to 
determine the motion of several bodies that mutually 
influence each other. This cannot be done without 
resorting to a new principle. Such a one Huygens 
actually discovered. 

We know that long pendulums perform their oscil- statement^ 
lations more slowly than short ones. Let us imagine a iem. 
heavy body, free to rotate about an axis, the centre of 
gravity of which lies outside of the axis; such " 
a body will represent a compound pendulum. 
Every material particle of a pendulum of this 
kind would, if it were situated alone at the 
same distance from the axis, have its own pe- 
riod of oscillation. But owing to the connec- Fig. 115. 
tions of the parts the whole body can vibrate with only 
a single, determinate period of oscillation. If we pic- 
ture to ourselves several pendulums of unequal lengths, 
the shorter ones will swing quicker, the longer ones 
slower. If all be joined together so as to form a single 
pendulum, it is to be presumed that the longer ones 
will be accelerated, the shorter ones retarded, and that 
a sort of mean time of oscillation will result. There 
must exist therefore a simple pendulum, intermediate 
in length between the shortest and the longest, that 
has the same time of oscillation as the compound pen- 
dulum. If we lay off the length of this pendulum on 
the compound pendulum, we shall find a point that pre- 
serves the same period of oscillation in its connection 



174 TIIE SCIENCE OF MECHANICS. 

with the other points as it would have if detached and 
left to itself. This point is the centre of oscillation. 
Mersenne was the first to propound the problem of 
determining the centre of oscillation. The solution of 
Descartes, who attempted it, was, however, precipi- 
tate and insufficient. 
Huygens's 17. Huygens was the first who gave a general solu- 

solution. . t->-itt 

tion. Besides Huygens nearly all the great inquirers 
of that time employed themselves on the problem, and 
we may say that the most important principles of mod- 
ern mechanics were developed in connection with it. 

The new idea from which Huygens set out, and 
which is more important by far than the whole prob- 
lem, is this. In whatsoever manner the material par- 
ticles of a pendulum may by mutual interaction modify 
each other's motions, in every case the velocities ac- 
quired in the descent of the pendulum can be such only 
that by virtue of them the centre of gravity of the par- 
ticles, whether still in connection or with their connec- 
tions dissolved, is able to rise just as high as the point 
The new from which it fell. Hu3'gens found himself compelled, 
whTch^Huy- by the doubts of his contemporaries as to the correct- 
duced n r ness of this principle, to remark, that the only assump- 
tion implied in the principle is, that heavy bodies of 
themselves do not move upwards. If it were possible 
for the centre of gravity of a connected system of falling 
material particles to rise higher after the dissolution 
of its connections than the point from which it had 
fallen, then by repeating the process heavy bodies 
could, by virtue of their own weights, be made to rise 
to any height we wished. If after the dissolution of 
the connections the centre of gravity should rise to a 
height less than that from which it had fallen, we 
should only have to reverse the motion to produce the 



THE PRIXCIPLES OF DYXAMICS. 175 

same result. What Huygens asserted, therefore, no 
one had ever really doubted ; on the contrary 7 , every 
one had instinctively perceived it. Huygens, however, 
gave this instinctive perception an abstract, conceptual 
form. He does not omit, moreover, to point out, on the 
ground of this view, the fruitlessness of endeavors to 
establish a perpetual motion. The principle just devel- 
oped will be recognised as a generalisation of one of Ga- 
lileo's ideas. 

18. Let us now see what the principle accomplishes Huygens's 

principle 

in the determination of the centre of oscillation. Let applied. 

Fig. 1 16), for simplicity's sake, 
be a linear pendulum, made up of a 
large number of masses indicated in 
the diagram by points. Set free at 
OA, it will swing through B to OA', 
where AB = BA'. Its centre of 
gravity S will ascend just as high 
on the second side as it fell on the Flg ' II5 ' 

first. From this, so far, nothing would follow. But 
also, if we should suddenly, at the position OB, re- 
lease the individual masses from their connections, the 
masses could, by virtue of the velocities impressed on 
them by their connections, only attain the same height 
with respect to centre of gravity. If we arrest the free 
outward-swinging masses at the greatest heights they 
severally attain, the shorter pendulums will be found 
below the line OA' , the longer ones will have passed 
beyond it, but the centre of gravity of the system will 
be found on OA' in its former position. 

Now let us note that the enforced velocities are 
proportional to the distances from the axis ; therefore, 
one being given, all are determined, and the height of 
ascent of the centre of gravity given. Conversely, 




176 



THE SC/EXCE OF MECHANICS. 



therefore, the velocity of any material particle also is 
determined by the known height of the centre of grav- 
ity. B.it if we know in a pendulum the velocity cor- 
responding to a given distance of descent, we know its 
whole motion. 
Thede- 19. Premising these remarks, we proceed to the 

tailed reso- 

lution of the problem itself. On a compound linear pendulum (Fig. 

problem. r . . 

117) we cut off, measuring from the axis, the 
portion = 1. If the pendulum move from its 
position of greatest excursion to the position 
of equilibrium, the point at the distance = 1 
from the axis will fall through the height k. 
The masses ;//, m', m" ... at the distances 
r, r', r" . . . will fall in this case the dis- 
tances rk, r' k, r" k . . . , and the distance of 
the descent of the centre of gravity will be : 




Fig. 117- 



m rk + m'r'k + m"r"k + . 



2mr 



in -j- m' -f- m" -f- . . . . 2 m 

Let the point at the distance 1 from the axis ac 
quire, on passing through the position of equilibrium 
the velocity, as yet unascertained, v. The height c 
its ascent, after the dissolution of its connections, wi , 
be v 2 /2g. The corresponding heights of ascent c 
the other material particles will then be (rv) 2 /2}.\ 
(V v) 2 /2g, (r" v) 2 /2g. . . . The height of ascent of th 
centre of gravity of the liberated masses will be 

{ rv Y 1 ,, (r'v} 2 ! \,, (r"z>) 2 

2 0- 2 m 



m 



2g 



-z 



&g 



m -j- *' -+- in" -(- -S 

By Huygens's fundamental principle, then, 
> 2 2mr 2 



2mr 

k 
2, m 



2 m 



(a). 



THE PRINCIPLES OF DYNAMICS. 177 

From this a relation is deducible between the distance of 
descent k and the velocity v. Since, however, all pen- 
dulum motions of the same excursion are phoronomi- 
cally similar, the motion here under consideration is, 
in this result, completely determined. 

To find the length of the simple pendulum that has The length 

b f f orthesim- 

the same period of oscillation as the compound pen- pie isoch- 
ronous 

dulum considered, be it noted that the same relation pendulum, 
must obtain between the distance of its descent and its 
velocity, as in the case of its unimpeded fall. If y is 
the length of this pendulum, ky is the distance of its 
descent, and vy its velocity ; wherefore 



*g 



= k v, or 



Multiplying equation (a) by equation (p) we obtain 

2 m r 2 

y = ^ . 

2. mr 

Employing the principle of phoronomic similitude, solution of 
we may also proceed in this way. From (a) we get iem by the 

principle of 
^ similitude. 

2 m r- 

A simple pendulum of length 1, under corresponding 
circumstances, has the velocity 



V2g*^ : 



* X -Vtgk. 

Calling the time of oscillation of the compound pendu- 
lum T, that of the simple pendulum of length 1 T t = 
~ 1 : g, we obtain, adhering to the supposition of 
equal excursions, 



T J 2mr 2 

-^-= ; wherefore T = tt a | ^ . 
T v \g2mr 



178 THE SCIENCE OF MECHANICS. 

Huygens's 20. We see without difficulty in the Huygenian 

principle .... . . 

identical principle the recognition of work as the condition de- 

withthe v . & 

principle oi terminative of velocity, or, more exactly, the condition 

vis viva. ... 

determinative of the so-called vis viva. By the vis 
viva or living force of a system of masses m, m 
m n . . . ., affected with the velocities v, v v n . . . ., we 
understand the sum * 

mv^ m,v 2 nt,,v, 2 

2 ^ 2 ~*~ 2 "*" 

The fundamental principle of Huygens is identical with 
the principle of vis viva. The additions of later in- 
quirers were made not so much to the idea as to the 
form of its expression. 

If we picture to ourselves generally any system of 
weights/, p,,p n . . . ., which fall connected or uncon- 
nected through the heights h, //,, //,,...., and attain 
thereby the velocities v, v,,v, t . . . ., then, by the Huy- 
genian conception, a relation of equality exists between 
the distance of descent and the distance of ascent of the 
centre of gravity of the system, and, consequently, the 
equation holds 

v 2 ,v' 2 v" 2 

P h + p ' h'+ p" h"+... = p 2 g + p 2 g +p "27 +; 
/+/+/' + /+/ + />" + 

1 t)V 2 

If we have reached the concept of "mass," which 
Huygens did not yet possess in his investigations, we 
may substitute for p/g the mass m and thus obtain th< 
form 2ph= %2 mv 2 , which is very easily generalised, 
for non-constant forces. 

* This is not the usual definition of English writers, who follow the olde 
authorities in making the vis viva twice this quantity. Trans. 




THE PRINCIPLES OF DYNAMICS. 179 

21. With the aid of the principle of living forces General 

r method of 

we can determine the duration of the infinitely small determin- 
ing the pe- 

oscillations of any pendulum whatso- / ^^ riodofpen- 

/ ^ dulum os- 

ever. We let fall from the centre of ^\ ciliations. 

gravity s (Fig. 1 18) a perpendicular on 
the axis ; the length of the perpendic- 
ular is, say, a. We lay off on this, 
measuring from the axis, the length 
= 1. Let the distance of descent of 
the point in question to the position of Flg - II8, 

equilibrium be k, and v the velocity acquired. Since 
the work done in the descent is determined by the 
motion of the centre of gravity, we have 

work done in descent = vis viva : 

akgM r=L 2 mr 2 . 

M here we call the total mass of the pendulum and 
anticipate the expression vis viva. By an inference 
similar to that in the preceding case, we obtain T= 
7t l/ 2 m r 2 J a gM. 

22. We see that the duration of infinitelv small The two 

- * determina- 

oscillations of any pendulum is determined by two fac- tive factors, 
tors by the value of the expression 2mr 2 , which 
Euler called the moment of inertia and which Huygens 
had employed without any particular designation, and 
by the value of agM. The latter expression, which we 
shall briefly term the statical moment, is the product 
a P of the weight of the pendulum into the distance of 
its centre of gravity from the axis. If these two values 
be given, the length of the simple pendulum of the 
same period of oscillation (the isochronous pendulum) 
and the position of the centre of oscillation are deter- 
mined. 



i8o 



THE SCIENCE OF MECHANICS. 



Huygens's For the determination of the lengths of the pendu- 

geometrical 

methods of lums referred to, Huygens, in the lack of the analytical 

solution. . 

methods later discovered, employed a very ingenious 
geometrical procedure, which 
we shall illustrate by one or 
two examples. Let the prob- 
lem be to determine the time 
of oscillation of a homogene- 
ous, material, and heavy rec- 
tangle ABCD, which swings 
on the axis AB (Fig. 119). 
Dividing the rectangle into 
minute elements of area/,/,, 
/,,.. . . . having the distances 
r, r,, r n . . . . from the axis, the expression for the 
length of the isochronous simple pendulum, or the dis- 
tance of the centre of oscillation from the axis, is given 
by the equation 




Fig. 119. 



/>-+/,>-,+/>+.. 

Let us erect on ABCD at C and D the perpendiculars 
C = >B=AC = BD and picture to ourselves a 
homogeneous wedge ABCDEF. Now find the distance 
of the centre of gravity of this wedge from the plane 
through AB parallel to CDEF. We have to consider, 
in so doing, the tiny columns fr,f, rf n r n . . . . and 
their distances r, r r n . . . . from the plane referred 
to. Thus proceeding, we obtain for the required dis- 
tance of the centre of gravity the expression 
fr r +f, r, . r, -f/ r . r + . . . . 

fr+f,r, +f,,r + 

that is, the same expression as before. The centre of 
oscillation of the rectangle and the centre of gravity of 



THE PRIXCIPLES OF DYXAMICS. 



181 



the wedge are consequently at the same distance from 
the axis, \AC. 

Following out this idea, we readily perceive the Analogous 

. applica- 

correctness of the following assertions. For a homo- tions of the 

preceding 

geneous rectangle of height h swinging about one of methods, 
its sides, the distance of the centre of gravity from the 
axis is hli, the distance of the centre of oscillation f //. 
For a homogeneous triangle of height //, the axis of 
which passes through the vertex parallel to the base, 
the distance of the centre of gravity from the axis is 
| h, the distance of the centre of oscillation |//. Call- 
ing the moments of inertia of the rectangle and of the 
triangle A lt A 2 , and their respective masses M v M 2 , 

we get 

A n 



3 h 



2*t 



Consequently A x = 



H 



k*M, 



T M 2 



^2 = 



/t*Af 9 



By this pretty geometrical conception many prob- 
lems can be solved that are to-day treated more con- 
veniently it is true by routine forms. 





Fig. 120. 



Fig. i2i. 



23. We shall now discuss a proposition relating to 
moments of inertia, that Huygens made use of in a 
somewhat different form. Let O (Fig. 121) be the 
centre of gravity of any given body. Make this the 



182 THE SCIENCE OF MECHANICS. 

The reia- origin of a system of rectangular coordinates, and sup- 

tion of mo- i 

ments of in- pose the moment of inertia with reference to the Z-axis 

ertia re- 
ferred to determined. If m is the element of mass and r its dis- 

parallel 

axes. tance from the Z-axis, then this moment of inertia is 

A = 2mr 2 . We now displace the axis of rotation 
parallel to itself to O', the distance a in the Jf-direction. 
The distance r is transformed, by this displacement, 
into the new distance p, and the new moment of 
inertia is 

@=2mp 2 = 2 m [(# a) 2 -f _>' 2 ] = 2m{x 2 -f y 2 ) 
2a2mx-\- a 2 2m, or, since 2m (x 2 -\-y 2 ) = 2mr 2 A, 
calling the total mass M= 2 m, and remembering the 
property of the centre of gravity 2 mx = 0, 

= ^ + a 2 M. 

From the moment of inertia for one axis through the 
centre of gravity, therefore, that for any other axis 
parallel to the first is easily derivable. 
An appii- 24. An additional observation presents itself here. 

cation of , x _ 

thispropo- The distance of the centre of oscillation is given bv 

sition. J 

the equation I = A -\- a 2 M/aM, where A, M, and a 
have their previous significance. The quantities A and 
M are invariable for any one given body. So long 
therefore as a retains the same value, / will also remain 
invariable. For all parallel axes situated at the same 
distance from the centre of gravity, the same body as 
pendulum has the same period of oscillation. If we 
put A/M x, then 

/= H - + a . 
a 

Now since / denotes the distance of the centre of 
oscillation, and a the distance of the centre of gravity 
from the axis, therefore the centre of oscillation is 
always farther away from the axis than the centre of 



THE PRIXCIPLES OF D YNAMICS. 183 

gravity by the distance x/a. Therefore x/a is the dis- 
tance of the centre of oscillation from the centre of 
gravity. If through the centre of oscillation we place 
a second axis parallel to the original axis, a passes 
thereby into x/a, and we obtain the new pendulum 
length 

x a a 

a 

The time of oscillation remains the same therefore 
for the second parallel axis through the centre of oscil- 
lation, and consequently the same also for every par- 
allel axis that is at the same distance x/a from the 
centre of gravity as the centre of oscillation. 

The totality of all parallel axes corresponding to 
the same period of oscillation and having the distances a 
and x/a from the centre of gravity, is consequently re- 
alised in two coaxial cylinders. Each generating line 
is interchangeable as axis with every other generating 
line without affecting the period of oscillation. 

25. To obtain a clear view of the relations subsist- The axial 
ing between the two axial cylinders, as we shall briefly 
call them, let us institute the following considerations. 
We put A = k % M t and then 

k- 

If we seek the a that corresponds to a given /, and 
therefore to a given time of oscillation, we obtain 
/ 17* 

Generally therefore to one value of / there correspond 
two values of a. Only where l / 2 4 k 2 = 0, that is 
in cases in which /= 2 k, do both values coincide in 
a = 6. 



184 



THE SCIENCE OF MECHANICS. 



If we designate the two values of a that correspond 
to every /, by a and ft, then 

/' 2 + /? 2 



/== 



k* + a'* 



or 



a ft 

/?(* + *)==(* + /?*)< 

k*(fta) = aft{ft a), 
k* = a . 0. 

The deter- If, therefore, in any pendulous body we know two par- 

mination of J *. * r 

the prece, 1 - allel axes that have the same time of oscillation and 

ing factor'- .#.. 

byageo- different distances a and ti from the centre of gravity, 

me-rical . . . 

method. as is the case for instance where we are able to give the 
centre of oscillation for any point of suspension, we 
can construct k. We lay off (Fig. 122) a and /? con- 




Fig. 122 



secutively on a straight line, describe a semicircle on 
a -f- ft as diameter, and erect a perpendicular at the 
point of junction of the two divisions a and ft. On this 
perpendicular the semicircle cuts off k. If on the other 
hand we know k, then for every value of a, say A, a 
value ju is obtainable that will give the same period 
of oscillation as A. We construct (Fig. 123) with A 
and k as sides a right angle, join their extremities by a 
straight line on which we erect at the extremity of k a 
perpendicular which cuts off on A produced the por- 
tion fA. 

Now let us imagine any body whatsoever (Fig. 124) 
with the centre of gravity 0. We place it in the plane 



THE PKIXCIPLES OF D YNAMICS. 185 

of the drawing, and make it swing about all possible An niustra- 

11 1 -i 1 11 r i tionofthis 

parallel axes at right angles to the plane of the paper, idea. 
All the axes that pass through the circle a are, we 
find, with respect to period of oscillation, interchange- 
able with each other and also with those that pass 
through the circle ft. If instead of a we take a smaller 
circle A, then in the place of ft we shall get a larger 




Fig. 124. 

circle fj. Continuing in this manner, both circles ul- 
timately meet in one with the radius k. 

26. We have dwelt at such length on the foregoing Recapituia 
matters for good reasons. In the first place, they have 
served our purpose of displaying in a clear light the 
splendid results of the investigations of Huygens. For 
all that we have given is virtually contained, though 
in somewhat different form, in the writings of Huygens, 



186 THE SCIENCE OE MECHANICS. 

or is at least so approximately presented in them that 
it can be supplied without the slightest difficulty. Only 
a very small portion of it has found its way into our 
modern elementary text-books. One of the proposi- 
tions that has thus been incorporated in our elemen- 
tary treatises is that referring to the convertibility of 
the point of suspension and the centre of oscillation. 
The usual presentation, however, is not exhaustive. 
Captain Kater, as we know, employed this principle 
for determining the exact length of the seconds pen- 
dulum. 
Function of The points raised in the preceding paragraphs have 

the moment . o jr o r 

of inertia, also rendered us the service of supplying enlighten- 
ment as to the nature of the conception "moment of 
inertia." This notion affords us no insight, in point 
of principle, that we could not have obtained without 
it. But since we save by its aid the individual con- 
sideration of the particles that make up a system, or 
dispose of them once for all, we arrive by a shorter 
and easier way at our goal. This idea, therefore, has 
a high import in the economy of mechanics. Poinsot, 
after Euler and Segner had attempted a similar object 
with less success, further developed the ideas that be- 
long to this subject, and by his ellipsoid of inertia and 
central ellipsoid introduced further simplifications. 
The lesser 27. The investigations of Huygens concerning the 
tions of geometrical and mechanical properties of cycloids are 
of less importance. The cycloidal pendulum, a contriv- 
ance in which Huygens realised, not an approximate, 
but an exact independence of the time and amplitude 
of oscillation, has been dropt from the practice of mod- 
ern horology as unnecessary. We shall not, therefore, 
enter into these investigations here, however much of 
the geometrically beautiful they may present. 



THE PRINCIPLES OF DYNAMICS. 187 

Great as the merits of Huygens are with respect to Huygens's 

1-rr i'ii 1 r 1 crowning 

the most different physical theories, the art of horology, achieve- 
practical dioptrics, and mechanics in particular, his 
chief performance, the one that demanded the greatest 
intellectual courage, and that was also accompanied 
with the greatest results, remains his enunciation of the 
principle by which he solved the problem of the centre 
of oscillation. This very principle, however, was the 
only one he enunciated that was not adequately appre- 
ciated by his contemporaries; nor was it for a long 
period thereafter. We hope to have placed this prin- 
ciple here in its right light as identical with the prin- 
ciple of vis viva. (See Appendix, XIII., p. 530.) 



THE ACHIEVEMENTS OF NEWTON. 

i. The merits of Newton with respect to our sub- Newton's 
ject were twofold. First, he greatly extended the range 
of mechanical physics by his discovery of universal 
gravitation. Second, he completed the formal enunciation 
of the mechanical principles now generally accepted. Since 
his time no essentially new principle has been stated. 
All that has been accomplished in mechanics since his 
day, has been a deductive, formal, and mathematical 
development of mechanics on the basis of Newton's 
laws. 

2. Let us first cast a glance at Newton's achieve- His great 
mem in the domain of physics. Kepler had deduced discovery, 
from the observations of Tycho Brahe and his own, 
three empirical laws for the motion of the planets 
about the sun, which Newton by his new view rendered 
intelligible. The laws of Kepler are as follows : 

1) The planets move about the sun in ellipses, in 
one focus of which the sun is situated. 



1 88 THE SCIENCE OF MECHANICS. 

Kepler's 2) The radius vector joining each planet with the 

laws. Their . 

part in the sun describes equal areas in equal times. /#4^H 

discovery. V 

3) The cubes of the moan dictan ces of the planets 
from the sun are proportional to the squares of 
their times of revolution. 
He who clearly understands the doctrine of Galileo 
and Huygens, must see that a curvilinear motion im- 
plies deflective acceleration. Hence, to explain the phe- 
nomena of planetary motion, an acceleration must be 
supposed constantly directed towards the concave side 
of the planetary orbits. 
Central ac- Now Kepler's second law, the law of areas, is ex- 

celeration 

explains plained at once by the assumption of a constant plane- 

Kepler's . 

second law. tary acceleration towards the sun ; or rather, this ac- 
celeration is another form of expression for the same 
fact. If a radius vector describes 
in an element of time the area 
ABS (Fig. 125), then in the next 
equal element of time, assuming 
no acceleration, the area BCS 
will be described, where BC = 
AB and lies in the prolongation 
Fig- "5- of AB. But if the central accel- 

eration during the first element of time produces a 
velocity by virtue of which the distance BD will be 
traversed in the same interval, the next-succeeding 
area swept out is not BCS, but BBS, where CE is par- 
allel and equal to BD. But it is evident that BBS = 
BCS= ABS. Consequently, the law of the areas con- 
stitutes, in another aspect, a central acceleration. 

Having thus ascertained the fact of a central accel- 
eration, the third law leads us to the discovery of its 
character. Since the planets move in ellipses slightly 
different from circles, we may assume, for the sake of 




THE PRIXCIPLES OF DYNAMICS. 189 

simplicity, that their orbits actually are circles. If^, The formal 

._, . character 

it,, R 3 are the radii and T x , T 2 , T 3 the respective of this ac- 
times of revolution of the planets, Kepler's third law deducibie 

. from Kep- 

may be written as follows : ler's third 

R 3 R s R 3 

T^r = ^ = -~^ = . . = a constant. 

But we know that the expression for the central accel- 
eration of motion in a circle is <p=z \R it 2 j T 2 , or 
T 1 = 4 t 2 R/cp. Substituting this value we get 
(p 1 R 1 2 = <p R 2 2 = <p 3 R 3 2 = constant ; or 
(p = constant R 2 ; 
that is to say, on the assumption of a central accelera- 
tion inversely proportional to the square of the distance, 
we get, from the known laws of central motion, Kep- 
ler's third law ; and vice versa. 

Moreover, though the demonstration is not easily 
put in an elementary form, when the idea of a central 
acceleration inversely proportional to the square of the 
distance has been reached, the demonstration that this 
acceleration is another expression for the motion in 
conic sections, of which the planetary motion in ellipses 
is a particular case, is a mere affair of mathematical 
analysis. 

3. But in addition to the intellectual performanceTheqnes- 
just discussed, the way to which was fully prepared by physical 

t . . . characterof 

Kepler, Galileo, and Huygens, still another achieve- this accei- 

r -v.- 1 ii-i- eration. 

ment 01 .Newton remains to be estimated which in no 
respect should be underrated. This is an achievement 
of the imagination. We have, indeed, no hesitation 
in saying that this last is the most important of all. 
Of what nature is the acceleration that conditions the 
curvilinear motion of the planets about the sun, and 
of the satellites about the planets ? 



igo THE SCIENCE OF MECHANICS. 

The steps Newton perceived, with great audacity of thought, 

which orig- . "* 

inaiiyied and first in the instance of the moon, that this accel- 

Newton to . 

the idea of eration differed in no substantial respect from the ac- 

universal . , _ 

gravitation, celeration of gravity so familiar to us. It was prob- 
ably the principle of continuity, which accomplished 
so much in Galileo's case, that led him to his dis- 
covery. He was wont and this habit appears to be 
common to all truly great investigators to adhere as 
closely as possible, even in cases presenting altered 
conditions, to a conception once formed, to preserve 
the same uniformity in his conceptions that nature! 
teaches us to see in her processes. That which is a, 
property of nature at any one time and in any one 
place, constantly and everywhere recurs, though it 
may not be with the same prominence. If the attrac- 
tion of gravity is observed to prevail, not only on the 
surface of the earth, but also on high mountains and in 
deep mines, the physical inquirer, accustomed to con- 
tinuity in his beliefs, conceives this attraction as also 
operative at greater heights and depths than those ac- 
cessible to us. He asks himself, Where lies the limit 
of this action of terrestrial gravity ? Should its action 
not extend to the moon ? With this question the great 
flight of fancy was taken, of which, with Newton's in- 
tellectual genius, the great scientific achievement was 
but a necessary consequence. (See p. 531.) 
The appii- Newton discovered first in the case of the moon that 
this idea to the same acceleration that controls the descent of a 

the motion , . . , , 

ot the moon, stone also prevented this heavenly body from moving j 
away in a rectilinear path from the earth, and that, on I 
the other hand, its tangential velocity prevented it from ; 
falling towards the earth. The motion of the moon j 
thus suddenly appeared to him in an entirely new light. 
but withal under quite familiar points of view. The 



THE PRrXCIPLES OF DYXAMICS. 191 

ne^v conception was attractive in that it embraced ob- 
jects that previously were very remote, and it was con- 
vincing in that it involved the most familiar elements. 
This explains its prompt application in other fields and 
the sweeping character of its results. 

Newton not only solved by his new conception the its univer- 
sal applica- 
thousand years' puzzle of the planetary system, buttiontoaii 



matter. 



also furnished by it the key to the explanation of a 
number of other important phenomena. In the same| 
way that the acceleration due to terrestrial gravity ex- 
tends to the moon and to all other parts of space, so do 
the accelerations that are due to the other heavenly 
bodies, to which we must, by the principle of contin- 
uity, ascribe the same properties, extend to all parts 
of space, including also the earth. But if gravitation is 
not peculiar to the earth, its seat is not exclusively in the 
centre of the earth. Every portion of the earth, how- 
ever small, shares it. Every part of the earth attracts, 
or determines an acceleration of, every other part. 
Thus an amplitude and freedom of physical view were 
reached of which men had no conception previously to 
Newton's lime. 

A long series of propositions respecting the action The sweep 
of spheres on other bodies situated beyond, upon, or ter of its re- 
within the spheres ; inquiries as to the shape of the 
earth, especially concerning its flattening by rotation, 
sprang, as it were, spontaneously from this view. The 
riddle of the tides, the connection of which with the 
moon had long before been guessed, was suddenly ex- 
plained as due to the acceleration of the mobile masses 
of terrestrial water by the moon. (See p. 533.) 

4. The reaction of the new ideas on mechanics was 
a result which speedily followed. The greatly varying 
accelerations which by the new view the same bod}' be- 



192 THE SCIENCE OF MECHANICS. 

rhe effect came affected with according to its position in space, 

of the new . 

ideas on suggested at once the idea or variable weight, yet also 
pointed to one characteristic property of bodies which 
was constant. The notions of mass and weight were 
thus first clearly distinguished. The recognised vari- 
ability of acceleration led Newton to determine by spe- 
cial experiments the fact that the acceleration of gravity 
is independent of the chemical constitution of bodies; 
whereby new positions of vantage were gained for the 
elucidation of the relation of mass and weight, as will 
presently be shown more in detail. Finally, the uni- 
versal applicability of Galileo's idea of force was more 
palpably impressed on the mind by Newton's perform- 
ances than it ever had been before. People could no 
longer believe that this idea was alone applicable to the 
phenomenon of falling bodies and the processes most 
immediately connected therewith. The generalisation 
was effected as of itself, and without attracting partic- 
ular attention. 
Newton's 5. Let us now discuss, more in detail, the achieve- 

ments in ments of Newton as they bear upon the principles of 
ofmechan- mechanics. In so doing, we shall first devote" ourselves 
exclusively to Newton's ideas, seek to bring them for- \ 
cibly home to the reader's mind, and restrict our criti- 
cisms wholly to preparatory remarks, reserving the 
criticism of details for a subsequent section. On pe- 1 
rusing Newton's work {Philosophies Naturalis Principia 
Mathematica. London, 1687), the following things 
strike us at once as the chief advances beyond Galileo i 
and Huygens : 

1) The generalisation of the idea of force. 

2) The introduction of the concept of mass. 

3) The distinct and general formulation of the prin- 
ciple of the parallelogram of forces. 



THE PRIXCIPLES OF DYXAMICS. 193 

The statement of the law of action and reaction. 
6. With respect to the first point little is to be His attitude 

with regard 

added to what has already been said. Newton con- to the idea 

. - .0* force. 

ceives all circumstances determinative of motion, 
whether terrestrial gravity or attractions of planets, or 
the action of magnets, and so forth, as circumstances 
determinative of acceleration. He expressly remarks 
on this point that by the words attraction and the like 
he does not mean to put forward any theory concern- 
ing the cause or character of the mutual action referred 
to, but simply wishes to express (as modern writers 
say. in a differential form) what is otherwise expressed 
(that is, in an integrated form) in the description of the 
motion. Newton's reiterated and emphatic protesta- 
tions that he is not concerned with hypotheses as to the 
causes of phenomena, but has simply to do with the 
investigation and transformed statement of actual facts, 
a direction of thought that is distinctly and tersely 
uttered in his words "hypotheses non fingo," "I do 
not frame hypotheses," stamps him as a philosopher 
of the highest rank. He is not desirous to astound and The Regu- 

. , . .... . . . lae Philoso- 

startle, or to impress the imagination by the originality phandi. 
of his ideas : his aim is to know Nature * 

* This is conspicuously shown in the rales that Newton formed for the 
conduct of natural inquiry (the Regula Pkilosophandi) : 

" Rule I. No more causes of natural things are to be admitted than snch 
as truly exist and are sufficient to explain the phenomena of these things. 

" Rule II. Therefore, to natural effects of the same kind we must, as far 
as possible, assign the same causes ; e. g., to respiration in man and animals ; 
to the descent of stones in Europe and in America ; to the light of our kitchen 
fire and of the sun ; to the reflection of light on the earth and on the planets. 

" Rule III. Those qualities of bodies that can be neither increased nor 
diminished, and which are found to belong to all bodies within the reach of 
our experiments, are to be regarded as the universal qualities of all bodies. 
[Here follows the enumeration of the properties of bodies which has been in- 
corporated in all text-books.] 

" If it universally appear, by experiments and astronomical observations, 
that all bodies in the vicinity of the earth are heavy with respect to the earth, 
and this in proportion to the quantity of matter which they severally contain ; 



ig 4 



THE SCIENCE OF MECHANICS. 






^i 



The expe- 
riences 



The New- 7. With regard to the concept of " mass," it is to 

tonian con- 
cept of be observed that the formulation of Newton, which de- 
mass. 

fines mass to be the quantity of matter of a body as 

measured by the product of its volume and density, is 
unfortunate. As we can only define density as the mass 
of unit of volume, the circle is manifest. Newton felt 
distinctly that in every body there was inherent a prop- 
erty whereby the amount of its motion was determined, 
and perceived that this must be different from weight. 
He called it, as we still do, mass ; but he did not suc- 
ceed in correctly stating this perception. We shall re- 
vert later on to this point, and shall stop here only to 
make the following preliminary remarks. 

8. Numerous experiences, of which a sufficient nura- 
which point ber stood at Newton's disposal, point clearlv to the ex- 

to the exist- . * r J 

enceofsuch istence of a property distinct from weight, whereby the 

quantity of motion of the 

body to which it belongs is 

determined. If (Fig. 126) 

we tie a fly-wheel to a rope 

and attempt to lift it by 

means of a pulley, we feel 

the weight of the fly-wheel. 

If the wheel be placed 

on a perfectly cylindrical axle and well balanced, it j 

will no longer assume by virtue of its weight any de- 

terminate position. Nevertheless, we are sensible of 

that the moon is heavy with respect to the earth in the proportion of its mass, 
and our seas with respect to the moon ; and all the planets with respect to one 
another, and the comets also with respect to the sun ; we must, in conformity 
with this rule, declare, that all bodies are heavy with respect to one anotaer. 

"Rule IV. In experimental physics propositions collected by induction 
from phenomena are to be regarded either as accurately true or very nearly 
true, notwithstanding any contrary hypotheses, till other phenomena occur, by 
which they are made more accurate, or are rendered subject to exceptions. 

"This rule must be adhered to, that the results of induction may not be 
annulled by hypotheses." 




D 



THE PRIXCIPLES OF D YNAMICS. 195 

a powerful resistance the moment we endeavor to set Mass dir- 

. . tinct from 

the wheel in motion or attempt to stop it when in mo- weight, 
tion. This is the phenomenon that led to the enuncia- 
tion of a distinct property of matter termed inertia, or 
"force" of inertia a step which, as we have already 
seen, and shall further explain below is unnecessary. 
Two equal loads simultaneously raised, offer resistance 
by their weight. Tied to the extremities of a cord that 
passes over a pulley, they offer resistance to any mo- 
tion, or rather to any change of velocity of the pulley, 
by their mass. A large weight hung as a pendulum 
on a very long string can be held at an angle of slight 
deviation from the line of equilibrium with very little 
effort. The weight-component that forces the pendu- 
lum into the position of equilibrium, is very small, 
notwithstanding this we shall experience a con- 
siderable resistance if we suddenly attempt to move or 
stop the weight. A weight that is just supported by a 
balloon, although we have no longer to overcome its 
gravity, opposes a perceptible resistance to motion. 
Add to this the fact that the same body experiences in 
different geographical latitudes and in different parts 
of space very unequal gravitational accelerations and 
we shall clearly recognise that mass exists as a property 
wholly distinct from weight determining the amount of 
acceleration which a given force communicates to the 
bod)' to which it belongs. (See p. 536.) 

9. Important is Newton's demonstration that the Mass meas 

r . urable by 

mass of a body may, nevertheless, under certain con- weight, 
ditions, be measured by its weight. Let us suppose a 
body to rest on a support, on which it exerts by its weight 
a pressure. The obvious inference is that 2 or 3 such 
bodies, or one-half or one-third of such a body, will pro- 
duce a corresponding pressure 2, 3, , or ^ times as 



ig5 



THE SCIENCE OF MECHANICS. 



The prere- 
quisites of 
the meas- 
urement of 
mass by 
weight. 



Newton's 
establish- 
ment of 
these pre- 
requisites. 



mmmmm. 

Fig. 127. 



great. If we imagine the acceleration of descent in- 
creased, diminished, or wholly removed, we shall ex- 
pect that the pressure also will be increased, dimin- 
ished, or wholly removed. We thus see, that the pres- 
sure attributable to weight increases, decreases, and 
. . vanishes along with the ''quan- 

tity of matter" and the magni- 
tude of the acceleration of de- 
scent. In the simplest manner 
imaginable we conceive the pres- 
sure/ as quantitatively representable by the product of 
the quantity of matter m into the acceleration of descent 
g by / = mg. Suppose now we have two bodies that 
exert respectively the weight-pressures /, /', to which 
we ascribe the ' ' quantities of matter ' ' m, m', and which 
are subjected to the accelerations of descent g, g'; then 
p = mg and f = m' g'. If, now, we were able to prove, 
that, independently of the material (chemical) compo- 
sition of bodies, g=g' at every same point on the 
earth's surface, we should obtain //;//' =///'; that is 
to say, on the same spot of the earth's surface, it would 
be possible to measure mass by weight. 

Now Newton established this fact, that g is inde- 
pendent of the chemical composition of bodies, by 
experiments with pendulums of equal lengths but dif- 
ferent material, which exhibited equal times of oscilla- 
tion. He carefully allowed, in these experiments, for 
the disturbances due to the resistance of the air; this 
last factor being eliminated by constructing from differ- 
ent materials spherical pendulum-bobs of exactly the 
same size, the weights of which were equalised by ap- 
propriately hollowing the spheres. Accordingly, all 
bodies maybe regarded as affected with the same^, and 



THE PRINCIPLES OF DYNAMICS. 197 

their quantity of matter or mass can, as Newton pointed 
out. be measured by their weight. 

If we imagine a rigid partition placed between an Supple- 

rocntsrv 

assemblage of bodies and a magnet, the bodies, if the considera- 

.... tions. 

magnet be powerful enough, or at least the majority 
of the bodies, will exert a pressure on the partition. 
But it would occur to no one to employ this magnetic 
pressure, in the manner we employed pressure due to 
weight, as a measure of mass. The strikingly notice- 
able inequality of the accelerations produced in the 
different bodies b)' the magnet excludes any such idea. 
The reader will furthermore remark that this whole 
argument possesses an additional dubious feature, in 
that the concept of mass which up to this point has 
simply been named and felt as a necessity, hut not de- 
fined, is assumed by it. 

10. To Newton we owe the distinct formulation ofThedoc- 

r r i_ j tr ' ne f the 

the principle of the composition of forces.* If a bodycomposi- 
is simultaneously acted on by two forces (Fig. 128), forces, 
of which one would produce the 

r A 

motion AB and the other the 
motion AC in the same interval .*. 

of time, the body, since the two 
forces and the motions produced Ig ' I2 ' 

by them are independent of each other, will move in that 
interval of time to AD. This conception is in every 
respect natural, and distinctly characterises the essen- 
tial point involved. It contains none of the artificial 
and forced characters that were afterwards imported 
into the doctrine of the composition of forces. 

We may express the proposition in a somewhat 

* Roberval's (1668) achievements with respect to the doctrine of the com- 
position of forces are also to be mentioned here. Varignon and Lami have al- 
ready been referred to. (See the text, page 36.) 




i 9 8 THE SCIENCE OF MECHANICS. 

Discussion different manner, and thus bring it nearer its modern 

of the doc- . 

trine of the form. The accelerations that different forces impart 

composi- . . 

tion of to the same body are at the same time the measure of 

forces. 

these forces. But the paths described in equal times 
are proportional to the accelerations. Therefore the 
latter also may serve as the measure of the forces. We 
may say accordingly : If two forces, which are propor- 
tional to the lines AB and A C, act on a body A in the 
directions AB and AC, a motion will result that could 
also be produced by a third force acting alone in the 
direction of the diagonal of the parallelogram con- 
structed on AB and AC and proportional to that di- 
agonal. The latter force, therefore, may be substituted 
for the other two. Thus, if cp and ip are the two ac- 
celerations set up in the directions AB and AC, then 
for any definite interval of time /, AB = cpt 2 ji, AC = 
ip t 2 /2. If, now, we imagine^/? produced in the same 
interval of time by a single force determining the accel- 
eration x> we g et 

AD =X* 2 / 2 > andAB : AC : AD = <p : ip : J. 
As soon as we have perceived the fact that the forces are 
independent of each other, the principle of the paral- 
lelogram of forces is easily reached from Galileo's no- 
tion of force. Without the assumption of this inde- 
pendence any effort to arrive abstractly and philosoph- 
ically at the principle, is in vain. 
The law of ii. Perhaps the most important achievement ct 
reaction. Newton with respect to the principles is the distinct 
and general formulation of the law of the equality of 
action and reaction^ of pressure and counter-pressure. 
Questions respecting the motions of bodies that exert 
a reciprocal influence on each other, cannot be soived 
by Galileo's principles alone. A new principle is ne- 
cessary that will define this mutual action. Such a 



THE PRINCIPLES OF DYNAMICS. 199 

principle was that resorted to by Huygens in his inves- 
tigation of the centre of oscillation. Such a principle 
also is Newton's law of action and reaction. 

A body that presses or pulls another body is, ac- Newton's 

1 11 deduction 

cording to Newton, pressed or pulled in exactly the of the law 

' r r J of action 

same degree bv that other body. Pressure and counter- and reac- 



tion. 



pressure, force and counter-force, are always equal to 
each other. As the measure of force is defined by 
Newton to be the quantity of motion or momentum 
(mass X velocity ) generated in a unit of time, it conse- 
quently follows that bodies that act on each other com- 
municate to each other in equal intervals of time equal 
and opposite quantities of motion (momenta), or re- 
ceive contrary velocities reciprocally proportional to 
their masses. 

Now, although Newton's law, in the form here ex-Thereia- 

. tive imme- 

pressed, appears much more simple, more immediate, diacyof 
and at first glance more admissible than that of Huv- and Hu>- 

gens's prin- 

gens. it will be found that it by no means contains less cipies. 
unanalysed experience or fewer instinctive elements. 
Unquestionably the original incitation that prompted 
the enunciation of the principle was of a purely instinc- 
tive nature. We know that we do not experience any 
resistance from a body until we seek to set it in motion. 
The more swiftly we endeavor to hurl a heavy stone 
from us, the more our body is forced back by it. Pres- 
sure and counter-pressure go hand in hand. The as- 
sumption of the equality of pressure and counter-pres- 
sure is quite immediate if, using Newton's own illus- 
tration, we imagine a rope stretched between two bod- 
ies, or a distended or compressed spiral spring between 
them. 

There exist in the domain of statics very many in- 
stinctive perceptions that involve the equality of pres- 



200 THE SCIENCE OF MECHANICS. 

statical ex- sure and counter-pressure. The trivial experience that 

periences 

which point one cannot lift ones self by pulling on one's chair is 

to the exist- . , 

enceof the of this character. In a scholium in which he cites the 
physicists Wren, Huygens, and Wallis as his prede- 
cessors in the employment of the principle, Newton 
puts forward similar reflections. He imagines the 
earth, the single parts of which gravitate towards one 
another, divided by a plane. If the pressure of the 
one portion on the other were not equal to the counter- 
pressure, the earth would be compelled to move in the 
direction of the greater pressure. But the motion of 
a body can, so far as our experience goes, only be de- 
termined by other bodies external to it. Moreover, 
we might place the plane of division referred to at any 
point we chose, and the direction of the resulting mo- 
tion, therefore, could not be exactly determined. 

The con- 12. The indistinctness of the concept of mass takes 

ceptofmass 

in its con- a very palpable form when we attempt to emplov the 

nection . . * . . 

with this principle of the equality of action and reaction dynam- 
ically. Pressure and counter-pressure may be equal. 
But whence do we know that equal pressures generate 
velocities in the inverse ratio of the masses ? Newton, 
indeed, actually felt the necessity of an experimental 
corroboration of this principle. He cites in a scholium, 
in support of his proposition, Wren's experiments on 
impact, and made independent experiments himself. 
He enclosed in one sealed vessel a magnet and in an- 
other a piece of iron, placed both in a tub of water, 
and left them to their mutual action. The vessels ap- 
proached each other, collided, clung together, and af- 
terwards remained at rest. This result is proof of the 
equality of pressure and counter-pressure and of equal 
and opposite momenta (as we shall learn later on, 
when we come to discuss the laws of impact). 



THE PRIXCIPLES OF DYNAMICS. 201 

The reader has already felt that the various enunci- The merits 

snd defects 

ations of Newton with respect to mass and the prin- of Newton's 
ciple of reaction, hang consistently together, and that 
they support one another. The experiences that lie at 
their foundation are : the instinctive perception of the 
connection of pressure and counter-pressure ; the dis- 
cernment that bodies offer resistance to change of ve- 
locity independently of their weight, but proportion- 
ately thereto ; and the observation that bodies of greater 
weight receive under equal pressure smaller velocities. 
Newton's sense of what fundamental concepts and prin- 
ciples were required in mechanics was admirable. The 
form of his enunciations, however, as we shall later in- * 
dicate in detail, leaves much to be desired. But we have 
no right to underrate on this account the magnitude of 
his achievements ; for the difficulties he had to conquer 
were of a formidable kind, and he shunned them less 
than any other investigator. AdA%ll " of S* - ^)** )*"*/ 

IV. 

DISCUSSION AND ILLUSTRATION OF THE PRINCIPLE OF 
REACTION. 



i. We shall now devote ourselves a moment ex- The princi- 
clusively to the Newtonian ideas, and seek to bring the tion. 
principle of reaction more clearly home to our mind 



V 



M 



Fig. 129. Fig. 130. 

and feeling. If two masses (Fig. 129) M and m act on 
one another, they impart to each other, according 
to Newton, contrary velocities Fand v, which are in- 
versely proportional to their masses, so that 
MV+mv = 0. 



202 THE SCIENCE OF MECHANICS. 

General The appearance of greater evidence may be im- 

elucidation .... 

of the prin- parted to this principle by the following consideration. 

cipleof re- -.-.-__. 

action. We imagine first (Fig. 130) two absolutely equal bodies 
a, also absolutely alike in chemical constitution. We 
set these bodies opposite each other and put them in 
mutual action ; then, on the supposition that the in- 
fluences of any third body and of the spectator are ex- 
cluded, the communication of equal and contrary velo- 
cities in the direction of the line joining the bodies is 
the sole uniquely determined interaction. 

Now let us group together in A (Fig. 131)//? such 
bodies a, and put at B over against them m such 
bodies a. We have then before us bodies whose quan- 

m'a. 
a 



M 



Fig. 131. Fig. 132. 

tities of matter or masses bear to each other the pro- 
portion m : ni '. The distance between the groups we 
assume to be so great that we may neglect the exten- 
sion of the bodies. Let us regard now the accelera- 
tions a, that every two bodies a impart to each other, 
as independent of each other. Every part of A, then, 
will receive in consequence of the action of B the ac- 
celeration m'a, and every part of B in consequence of 
the action of A the acceleration m a accelerations 
which will therefore be inversely proportional to the 
masses. 

2. Let us picture to ourselves now a mass M (Fig. 
132) joined by some elastic connection with a mass ///, 
both masses made up of bodies a equal in all respects. 
Let the mass m receive from some external source an 
acceleration cp. At once a distortion of the connection 
is produced, by which on the one hand m is retarded 



THE PRINCIPLES OF D YNAMICS. 203 

and on the other M accelerated. When both masses The deduc 

. tion of the 

have begun to move with the same acceleration, all notion of 
further distortion of the connection ceases. If we call force." 
a the acceleration of M and (5 the diminution of the 
acceleration of m, then a = <p /?, where agreeably 
to what precedes a M = fim. From this follows 

a , <*M m( P 

<* + P =* a H = <p, or a = / . 

m M-\-m 

If we were to enter more exhaustively into the de- 
tails of this last occurrence, we should discover that 
the two masses, in addition to their motion of progres- 
sion, also generally perform with respect to each other 
motions of oscillation. If the connection on slight dis- 
tortion develop a powerful tension, it will be impos- 
sible for any great amplitude of vibration to be reached, 
and we may entirely neglect the oscillatory motions, 
as we actually have done. 

If the expression a= m<p/M -{- m, which deter- 
mines the acceleration of the entire system, be ex- 
amined, it will be seen that the product m q> plays a 
decisive part in its determination. Newton therefore 
invested this product of the mass into the acceleration 
imparted to it, with the name of "moving force." 
M -\- m, on the other hand, represents the entire mass 
of the rigid system. We obtain, accordingly, the accel- 
eration of any mass m on which the 
moving force / acts, from the expres- 
sion pjm' . 

3. To reach this result, it is not at 
all necessary that the two connected 
masses should act directly on each other in all their 
parts. We have, connected together, let us say, the 
three masses m 13 m 2 , m 3 , where m^ is supposed to act 



"'3 




m, 




'". 



2o 4 THE SCIENCE OF MECHANICS. 

a condition only on m 9 , and ; only on m . Let the mass ;/z, re- 

which does . , X 

not affect ceive from some external source the acceleration a/. 

the pre- 

vious re- In the distortion that follows, the 

suit. 

masses m 3 m 2 m. 

receive the accelerations -\- 8 -f- /3 -\- q> 

y a. 
Here all accelerations to the right are reckoned as 
positive, those to the left as negative, and it is obvious 
that the distortion ceases to increase 

when 6 = (3 y, d = cp a, 
where 8m 3 = ym 2 , am x = fim 2 . 
The resolution of these equations yields the com- 
mon acceleration that all the masses receive ; namely, 

$ = m x<P 

a result of exactly the same form as before. When 
therefore a magnet acts on a piece of iron which is 
joined to a piece of wood, we need not trouble our- 
selves about ascertaining what particles of the wood 
are distorted directly or indirectly (through other par- 
ticles of the wood) by the motion of the piece of iron. 
The considerations advanced will, in some meas- 
ure, perhaps, have contributed towards clearly impress- 
ing on us the great importance for mechanics of the 
Newtonian enunciations. They will also serve, in a 
i subsequent" place, to ren- 

der more readily obvious 
the defects of these enun- 
i ' ciations. 



r _ 



L% 



^ f 4. Let us now turn to 

Fig. 134. a few illustrative physical 

examples of the principle of reaction. We consider, 
say, a load L on a table T. The table is pressed by 



THE PRIXCIPLES OF DYXAM1CS. 205 

the load just so much, and so much only, as it in return Some phys- 
presses the load, that is prevents the same from falling, pies of the 
If/ is the weight, m the mass, and g the acceleration of reaction, 
of gravity, then by Newton's conception/ = mg. If 
the table be let fall vertically downwards with the ac- 
celeration of free descent g, all pressure on it ceases. 
We discover thus, that the pressure on the table is de- 
termined by the relative acceleration of the load with 
respect to the table. If the table fall or rise with the 
acceleration y, the pressure on it is respectively m (g 
y) and m (g -)- y). Be it noted, however, that no 
change of the relation is produced by a constant velocity 
of ascent or descent. The relative acceleration is de- 
terminative. 

Galileo knew this relation of things very well. TheTnepres- 
doctrine of the Aristotelians, that bodies of greater parts of fail 

. ing bodies. 

weight fall faster than bodies of less weight, he not only 
refuted by experiments, but cornered his adversaries 
by logical arguments. Heavy bollies fall faster than 
light bodies, the Aristotelians said, because the upper 
parts weigh down on the under parts and accelerate 
their descent. In that case, returned Galileo, a small 
body tied to a larger body must, if it possesses in se the 
property of less rapid descent, retard the larger. There- 
fore, a larger body falls more slowly than a smaller 
body. The entire fundamental assumption is wrong, 
Galileo says, because one portion of a. falling body can- 
not by its weight under any circumstances press an- 
other portion. 

A pendulum with the time of oscillation T= rr V l/g, a falling 
would acquire, if its axis received the downward accel- pen u um * 
eration y, the time of oscillation T= tiV If^Z y, 
and if let fall freely would acquire an infinite time of 
oscillation, that is, would cease to oscillate. 



2o6 



THE SCIENCE OF MECHANICS. 



The sensa- 
tion of fall 
lag. 



i?ni"; v\ ua( 



Pogeen- 
dorff's ap- 
paratus. 



We ourselves, when we jump or fall from an eleva- 
tion, experience a peculiar sensation, which must be 
due to the discontinuance of the gravitational pressure 
of the parts of our body on one another the blood, and 
so forth. A similar sensation, as if the ground were 
sinking beneath us, we should have on a smaller planet, 
to which we were suddenly transported. The sensation 
of msteet ascent, like that felt in an earthquake, 
would be produced on a larger planet. 

5. The conditions referred to are very beautifully 
illustrated by an apparatus (Fig. 135^) constructed 
by Poggendorff. A string loaded at both extremities 



% 



P+p 



\2P+p 



6p 



& 1 



7d 



^> 



D 



Fig. 135a. Fig. 135b. 

by a weight P (Fig. 135a) is passed over a pulley c, 
attached to the end of a scale-beam. A weight / is 
laid on one of the weights first mentioned and tied by 
a fine thread to the axis of the pulley. The pulley 
now supports the weight 2 P -j- /. Burning away the 
thread that holds the over-weight, a uniformly accel- 
erated motion begins with the acceleration y, with 
which P -\- p descends and P rises. The load on the 
pulley is thus lessened, as the turning of the scales in- 
dicates. The descending weight P\s counterbalanced 
by the rising weight P, while the added over-weight, 
instead of weighing/, now weighs (/>/g)(g y)- And 
since y = (p/2 P -|- p) g, we have now to regard the 
load on the pulley, not as /, but as/(2 Pli P-\-p)- The 



THE PRIXCIPLES OF DYXAMICS. 



207 



descending weight, only partially impeded in its motion 
of descent, exerts only a partial pressure on the pulley. 

We may vary the experiment. We pass a thread a variation 

1 r> i. of the last 

loaded at one extremity with the weight P over the experiment 
pulleys a, b, d, of the apparatus as indicated in Fig. 




Fig. 135c. 



135^., tie the unloaded extremity at m, and equilibrate 
the balance. If we pull on the string at m, this can- 
not directly affect the balance since the direction of the 
string passes exactly through its axis. But the side a 
immediately falls. The slackening of the string causes 
a to rise. An unaccelerated motion of the weights would 



208 



THE SCIENCE OF MECHANICS. 



The suspen- 
sion of mi- 



different 

specific 

gravity. 




Fig. 136. 



not disturb the equilibrium. But we cannot pass from 
rest to motion without acceleration. 

6. A phenomenon that strikes us at first glance is, 
nute bodies that minute bodies of greater or less specific gravity 

in liquids of .... 

than the liquid in which they are immersed, if suffi- 
ciently small, remain suspended a very long time in the 
liquid. We perceive at once that 
particles of this kind have to over- 
come the friction of the liquid. If the 
cube of Fig. 136 be divided into 8 
parts by the 3 sections indicated, 
and the parts be placed in a row, 
their mass and over-weight will re- 
main the same, but their cross-sec- 
tion and superficial area, with which the friction goes 
hand in hand, will be doubled. 
Do such Now, the opinion has at times been advanced with 

suspended 

particles af- respect to this phenomenon that suspended particles 
specific of the kind described have no influence on the specific 

gravities of . . ,. . . 

the support- gravity indicated by an areometer immersed in the 
! liquids? ?. . , J J 

liquid, because these particles are themselves areo- 
meters. But it will readily be seen that if the sus- 
pended particles rise or fall with constant velocity, as 
in the case of very small particles immediately occurs, 
the effect on the balance and the areometer must be 
the same. If we imagine the areometer to oscillate 
about its position of equilibrium, it will be evident 
that the liquid with all its contents will be moved with | 
it. Applying the principle of virtual displacements, 
therefore, we can be no longer in doubt that the areo- 
meter must indicate the mean specific gravity. We 
may convince ourselves of the untenability of the rule 
by which the areometer is supposed to indicate only 
the specific gravity of the liquid and not that of the sus- 



ing hqi 



THE PRIXCIPLES OF DYNAMICS. 209 

{ended particles, by the following consideration. In a 
liquid A a smaller quantity of a heavier liquid B is in- 
troduced and distributed in fine drops. The areometer, 
let us assume, indicates only the specific gravity of 
A. Now, take more and more of the liquid B, finally 
just as much of it as we have of A: we can, then, no 
longer say which liquid is suspended in the other, and 
which specific gravity, therefore, the areometer must 
indicate. 

7. A phenomenon of an imposing kind, in which The phe- 
the relative acceleration of the bodies concerned is the tides, 
seen to be determinative of their mutual pressure, is 
that of the tides. We will enter into this subject here 
only in so far as it may serve to illustrate the point we 
are considering. The connection of the phenomenon 
of the tides with the motion of the moon asserts itself 
in the coincidence of the tidal and lunar periods, in 
the augmentation of the tides at the full and new 
moons, in the daily retardation of the tides (by about 
50 minutes i, corresponding to the retardation of the 
culmination of the moon, and so forth. As a matter 
of fact, the connection of the two occurrences was very 
early thought of. In Newton's time people imagined 
to themselves a kind of wave of atmospheric pressure, 
by means of which the moon in its motion was sup- 
posed to create the tidal wave. 

The phenomenon of the tides makes, on every one its impos- 

.1 . . , - r .. . ingcharac- 

tnat sees it for the first time in its full proportions, an ter. 
overpowering impression. We must not be surprised, 
therefore, that it is a subject that has actively engaged 
the investigators of all times. The warriors of Alex- 
ander the Great had, from their Mediterranean homes, 
scarcely the faintest idea of the phenomenon of the 
' tides, and they were, therefore, not a little taken aback 



210 THE SCIENCE OF MECHANICS. 

by the sight of the powerful ebb and flow at the mouth 
of the Indus ; as we learn from the account of Curtius 
Rufus (De Rebus Gestis Alexandri Magni), whose 
words we here literally quote : 
Extract "34- Proceeding, now, somewhat more slowly in 

from Cur- . . J 

tius Rufus. "tneir course, owing to the current of the river being 
" slackened by its meeting the waters of the sea, they 
"at last reached a second island in the middle of the 
"river. Here they brought the vessels to the shore, 
"and, landing, dispersed to seek provisions, wholly 
"unconscious of the great misfortune that awaited j 
"them. 

Describing << 35. It was about the third hour, when the ocean.' 

the effect . . 

on the army "in its constant tidal flux and reflux, began to tun 

of Alexan- ' 

derthe "and press back upon the river. The latter, at firs 

Great of the r ' 

tides at the " merely checked, but then more vehemently repelled 

mouth of . ... 

the Jn<*u*. " at last set back in the opposite direction with a fora 
"greater than that of a rushing mountain torrent 
" The nature of the ocean was unknown to the multi 
" tude, and grave portents and evidences of the wratl 
"of the Gods were seen in what happened. Witl 
"ever- increasing vehemence the sea poured in, com; 
"pletely covering the fields which shortly before wen 
" dry. The vessels were lifted and the entire fleet di? 
" persed before those who had been set on shore, ter 
"rifled and dismayed at this unexpected calamit} 
"could return. But the more haste, in times of gi 
"disturbance, the less speed. Some pushed the ship 
" to the shore with poles ; others, not waiting to adjm 
"their oars, ran aground. Many, in their great hast 
"to get away, had not waited for their companion' 
"and were barely able to set in motion the huge, un 
" manageable barks ; while some of the ships were to 
"crowded to receive the multitudes that struggled t 



THE PRINCIPLES OF DYNAMICS. 211 

'get aboard. The unequal division impeded all. TheThedisas- 
" cries of some clamoring to be taken aboard, of others ander's 
^crying to put off, and the conflicting commands of 
'men, all desirous of different ends, deprived every one 
of the possibility of seeing or hearing. Even the 
-steersmen were powerless; for neither could their 

cries be heard by the struggling masses nor were their 
"orders noticed by the terrified and distracted crews. 

The vessels collided, they broke off each other's oars, 

' " they plunged against one another. One would think 

it was not the fleet of one and the same army that 

is here in motion, but two hostile fleets in combat. 

Prow struck stern ; those that had thrown the fore- 
"most in confusion were themselves thrown into con- 
'* fusion by those that followed; and the desperation 
-of the struggling mass sometimes culminated in 
'hand-to-hand combats. 

"36. Already the tide had overflown the fields sur- 

unding the banks of the river, till only the hillocks 

"jutted forth from above the water, like islands. 

! "These were the point towards which all that had given 

"up hope of being taken on the ships, swam. The 

'scattered vessels rested in part in deep water, where 

there were depressions in the land, and in part lay 

-.round in shallows, according as the waves had 

covered the unequal surface of the country. Then, 

"suddenly, a new and greater terror took possession 

"of them. The sea began to retreat, and its waters 

flowed back in great long swells, leaving the land 

"which shortly before had been immersed by the salt 

"waves, uncovered and clear. The ships, thus for- 

" saken by the water, fell, some on their prows, some 

"on their sides. The fields were strewn with luggage, 

"arms, and pieces of broken planks and oars. The 



212 THE SCIENCE OF MECHANICS. 

The dismay << soldiers dared neither to venture on the land nor to 

of the army. 

"remain in the ships, for every moment they expected 
"something new and worse than had yet befallen 
"them. They could scarcely believe that that which 
"they saw had really happened a shipwreck on dry 
"land, an ocean in a river. And of their misfortune 
"there seemed no end. For wholly ignorant that the 
"tide would shortly bring back the sea and again set 
" their vessels afloat, they prophesied hunger and dir- 
"est distress. On the fields horrible animals crept 
"about, which the subsiding floods had left behind. 
2?t e hlw? s " 37 ' Tne ni ght fell, and even the king was sore 
and the re- "distressed at the slight hope of rescue. But his so- 

turn ot the u * 

tide. " licitude could not move his unconquerable spirit. He 

"remained during the whole night on the watch, and 
"despatched horsemen to the mouth of the river, that, 
"as soon as they saw the sea turn and flow back, they 
"might return and announce its coming. He also 
"commanded that the damaged vessels should be re- 
" paired and that those that had been overturned by 
"the tide should be set upright, and ordered all to be 
" near at hand when the sea should again inundate the 
"land. After he had thus passed the entire night in 
"watching and in exhortation, the horsemen came { 
" back at full speed and the tide as quickly followed, j 
"At first, the approaching waters, creeping in light ' 
"swells beneath the ships, gently raised them, and, 
"inundating the fields, soon set the entire fleet in mo- ! 
"tion. The shores resounded with the cheers and 
"clappings of the soldiers and sailors, who celebrated 
" with immoderate joy their unexpected rescue. 'But 
" whence,' they asked, in wonderment, 'had the sea' 
" so suddenly given back these great masses of water? 
"Whither had they, on the day previous, retreated? 



THE PRIXCIPLES OF DYXAMICS. 



213 



" And what was the nature of this element, which now 
"opposed and now obeyed the dominion of the hours? ' 

- the king concluded from what had happened that 
"the fixed time for the return of the tide was after 
"sunrise, he set out, in order to anticipate it, at mid- 
" night, and proceeding down the river with a few 
"ships he passed the mouth and, finding himself at 
"last at the goal of his wishes, sailed out 400 stadia 
"into the ocean. He then offered a sacrifice to the 
"divinities of the sea, and returned to his fleet." 

8. The essential point to be noted in the explana- The expia- 

..,., ration of 

tion of the tides is, that the earth as a rigid body canthephe- 

, nomena of 

receive but one determinate acceleration towards the the tides, 
moon, while the mobile particles of water on the sides 
nearest to and remotest from the moon can acquire 
various accelerations. 




Fitf- 137- 

Let us consider (Fig. 137) on the earth E, opposite 
which stands the moon M, three points A, B, C. The 
accelerations of the three points in the direction of the 
moon, if we regard them as free points, are respect- 
ively cp -{- A cp, cp, cp A cp. The earth as a whole, 
however, has, as a rigid body, the acceleration <p. The 
acceleration towards the centre of the earth we will 
call g. Designating now all accelerations to the left 
as negative, and all to the right as positive, we get the 
following table : 



4 THE SCIEXCE OF MECHANICS. 

ABC 

(JP + ^90* ^ (f ^9) 
+ * F 

<P* <Pi 9> 

^ ^ 0, teJf), 

h::v :':.- ?;. r;:'r :^ :: :hr r.rs: in : st: :~ i - vr :r::r- 
sent the accelerations which the free points that h 
:!r.r ::.i~-s rt:e:ve. :/:-t :: :;.t ::.::: .:~t ::.t i::-.-.- 
e:;:::~ :: ; : rrts^: ~ i.r. _ r;;:i : :.~:- :: :_-. t ^r.:.. i:: 
:h:;t :: :': .-:- : : :r.r. ..--_-. :hr iirtrt- :e. :r :'- r 5 _ 1 : i - : 
accelerations of the free points towards the earth. It 
will be seen from this result that the weight of the v 
at A and C is diminished by exactly the same amou:: 
The water will rise at A and C (Fig- 137)- A : 
wave will be produced at these points twice e 
lay. 

It is a fact not always sufficiently emphasised, that 

:: :;.r -.:::. i:: ::\t v.,:::.- :v r. :: ir't-::-: - ::r. i:- 
celerated motion towards each other but were relath e 7 
fixed and at rest. If we modify the considers: 
presented to comprehend this case, we must put for 

:_ : v\:: :.:'..:- :-_ - . : :. 7 .::.: : :: := >:~p;y. 
We then obtain for 



the free points A 


C 


:hr i: ir.-rri:: :r^ _~ _ j- . 


V J& 


+ ^ 


g 



or > J<p) gK .- -'< ?> y 

or ** ?>, 

where g'=g -Jpi. In such case, therefore, 
weight of the water at A would be diminished, and 
weight at C increased ; the height of die water 2 



r. '- 1 ~:~::?li: :j z: -.:::: 



--! 







prik :i At 



216 THE SCIENCE OF MECHANICS. 

continuously along over the surface of the water of a 
circular canal, a wave of considerable magnitude fol- 
lowing in the wake of the fan would by this gentle and 
constantly continued impulsion soon be produced. In 
like manner the tide is produced. But in the latter 
case the occurrence is greatly complicated by the irreg- 
ular formation of the continents, by the periodical 
variation of the disturbance, and so forth. (See Ap- 
pendix, XVII., p. 537.) 



CRITICISM OF THE PRINCIPLE OF REACTION AND OF THE 
CONCEPT OF MASS. 

The con- I. Now that the preceding discussions have made 

mass us familiar with Newton's ideas, we are sufficiently 
prepared to enter on a critical examination of them. 
We shall restrict ourselves primarily in this, to the 
consideration of the concept of mass and the principle 
of reaction. The two cannot, in such an examination, 
be separated ; in them is contained the gist of New- 
ton's achievement. 
Theexpres- 2. In the first place we do not find the expression 

sion " quan- . r ,,i 1 1 j 1 j 1. 

tityof mat- "quantity of matter adapted to explain and elucidate 
the concept of mass, since that expression itself is not 
possessed of the requisite clearness. And this is so, 
though we go back, as many authors have done, to an 
enumeration of the hypothetical atoms. We only com- 
plicate, in so doing, indefensible conceptions. If we 
place together a number of equal, chemically homo- 
geneous bodies, we can, it may be granted, connect 
some clear idea with "quantity of matter," and we per- 
ceive, also, that the resistance the bodies offer to mo- 
tion increases with this quantity. But the moment we 
suppose chemical heterogeneity, the assumption that 



ter. 



THE PRIXCIPLES OF DYXAMICS. 217 

there is still something that is measurable by the same Newton's 

11 formulation 

standard, which something we call quantity of matter, of the con- 
cept, 
maybe suggested by mechanical experiences, but is an 

assumption nevertheless that needs to be justified. 
When therefore, with Newton, we make the assump- 
tions, respecting pressure due to weight, that/ = mg, 
p' m'g, and put in conformity with such assumptions 
p p' = mhri , we have made actual use in the operation 
thus performed of the supposition, yet to be justified, 
that different bodies are measurable by the same stand- 
ard. 

We might, indeed, arbitrarily posit, that mjin =plp'; 
that is, might define the ratio of mass to be the ratio 
of pressure due to weight when g was the same. But 
we should then have to substantiate the use that is made 
of this notion of mass in the principle of reaction and 
in other relations. 

- - <~ A *-> 

' ' cp -cp 

Fig. 140 a. Fig. 140 b. 

3. When two bodies (Fig. 140 a), perfectly equal a new form- 
in all respects, are placed opposite each other, we ex- the con- 
pect, agreeably to the principle of symmetry, that they 
will produce in each other in the direction of their line 
of junction equal and opposite accelerations. But if 
these bodies exhibit any difference, however slight, of 
form, of chemical constitution, or are in any other re- 
spects different, the principle of symmetry forsakes us, 
unless we assume or know beforehand that sameness of 
form or sameness of chemical constitution, or whatever 
else the thing in question may be, is not determina- 
tive. If, however, mechanical experiences clearly and 
indubitably point to the existence in bodies of a special 
and distinct property determinative of accelerations, 



2i8 THE SCIENCE OF MECHANICS. 

nothing stands in the way of our arbitrarily establish- 
ing the following definition : 
Definition All those bodies are bodies of equal mass, which, mu- 

masses. tually acting on each other, produce in each other equal 
and opposite accelerations. 

We have, in this, simply designated, or named, an 
actual relation of things. In the general case we pro- 
ceed similarly. The bodies A and B receive respec- 
tively as the result of their mutual action (Fig. 140 b) 
the accelerations cp and -)- cp' , where the senses of 
the accelerations are indicated by the signs. We say 
then, B has cp/cp' times the mass of A. If we take A 
as our unit, we assign to that body the mass m which im- 
parts to A m times the acceleration that A in the reaction 
imparts to- it. The ratio of the masses is the negative 
inverse ratio of the counter-accelerations. That these 
accelerations always have opposite signs, that there 
are therefore, by our definition, only positive masses, 
is a point that experience teaches, and experience alone 
character can teach. In our concept of mass no theory is in- 
nition. volved ; " quantity of matter " is wholly unnecessary in 
it ; all it contains is the exact establishment, designa- 
tion, and denomination of a fact. (Compare Appendix, 
XVIIL, p. 539.) 

4. One difficulty should not remain unmentioned in 
this connection, inasmuch as its removal is absolutely 
necessary to the formation of a perfectly clear concept 
of mass. We consider a set of bodies, A, B, C, D . . ., 
and compare them all with A as unit. 



A, 


B, 


c, 


D, 


E, 


F. 


1, 


m, 


m , 


m" , 


m'" , 


m" 



We find thus the respective mass-values, 1, m, ta' , 
m". . . ., and so forth. The question now arises, If we 



THE PRINCIPLES OF DYNAMICS. 219 

select B as our standard of comparison (as our unit), Discussion 
shall we obtain for C the mass-value m'/m, and for D cuity in- 
the value m" ]m, or will perhaps wholly different values the preced- 
result? More simply, the question maybe put thus : lation. 
Will two bodies B, C, which in mutual action with A 
have acted as equal masses, also act as equal masses 
in mutual action with each other? No logical necessity 
exists whatsoever, that two masses that are equal to a 
third mass should also be equal to each other. For 
we are concerned here, not with a mathematical, but 
with a physical question. This will be rendered quite 
clear by recourse to an analogous relation. We place 
by the side of each other the bodies A, B, C in the 
proportions of weight a, b, c in which they enter into 
the chemical combinations AB and AC. There exists, 
now, no logical necessity at all for assuming that the 
same proportions of weight b, c of the bodies B, C will 
also enter into the chemical combination BC. Expe- 
rience, however, informs us that they do. If we place 
by the side of each other any set of bodies in the pro- 
portions of weight in which they combine with the 
body A, they will also unite with each other in the 
same proportions of weight. But no one can know 
this who has not tried it. And this is precisely the case 
with the mass-values of bodies. 

If we were to assume that the order of combination The order 

..... ..,,. , . of combi- 

ot the bodies, by which their mass-values are deter- nation not 
mined, exerted any influence on the mass-values, the 
consequences of such an assumption would, we should 
find, lead to conflict with experience. Let us suppose, 
for instance (Fig. 141), that we have three elastic 
bodies, A, B, C, movable on an absolutely smooth and 
rigid ring. We presuppose that A and B in their 
mutual relations comport themselves like equal masses 




220 THE SCIENCE OF MECHANICS. 

and that B and C do the same. We are then also 
obliged to assume, if we wish to avoid conflicts with 
experience, that C and A in their mutual relations act 
like equal masses. If we impart to A a velocity, A 
will transmit this velocity by impact to B, and B to C. 
But if C were to act towards A, say, as a greater mass, 
A on impact would acquire a greater 
velocity than it originally had while 
C would still retain a residue of 
what it had. With every revolution 
in the direction of the hands of a 
watch the vis viva of the system 
would be increased. If C were the 
lg ' I41 ' smaller mass as compared with A, 

reversing the motion would produce the same result. 
But a constant increase of vis viva of this kind is at 
decided variance with our experience. 
The new 5. The concept of mass when reached in the man- 

massm- ner just developed renders unnecessary the special 
plicitly the enunciation of the principle of reaction. In the con- 
reaction 6 cept of mass and the principle of reaction, as we have 
stated in a preceding page, the same fact is twice form- 
ulated; which is redundant. If two masses i and 2 
act on each other, our very definition of mass asserts 
that they impart to each other contrary accelerations 
which are to each other respectively as 2 : i. 

6. The fact that mass can be measured by weight, 
where the acceleration of gravity is invariable, can also 
be deduced from our definition of mass. We are 
sensible at once of any increase or diminution of a pres- 
sure, but this feeling affords us only a very inexact and 
indefinite measure of magnitudes of pressure. An 
exact, serviceable measure of pressure springs from 
the observation that every pressure is replaceable by 



THE PRIXCIPLES OF DYNAMICS. 221 

the pressure of a number of like and commensurable it also in- 

volves the 

weights. Every pressure can be counterbalanced by fact that 

" . ' mass can be 

the pressure of weights of this kind. Let two bodies measured 

. . by weight. 

ni and ///' be respectively affected in opposite directions 
with the accelerations cp and cp' , determined by exter- 
nal circumstances. And let the bodies be joined by a 
string. If equilibrium prevails, the acceleration cp in 
m and the acceleration cp' in m' are exactly balanced 
by interaction. For this case, ac- 
cordingly, /ncp = ;n'cp'. When, < 
therefore, cp = cp', as is the case ^ 



9 



when the bodies are abandoned 

to the acceleration of gravity, we have, in the case 
of equilibrium, also m = in . It is obviously imma- 
terial whether we make the bodies act on each other 
directly by means of a string, or by means of a string 
passed over a pulley, or by placing them on the two 
pans of a balance. The fact that mass can be meas- 
ured by weight is evident from our definition without 
recourse or reference to "quantity of matter." 

7. As soon therefore as we, our attention being The general 

j 1 r 1 i 7 i_ j results of 

drawn to the tact by experience, have perceived in bod- this view, 
ies the existence of a special property determinative of 
accelerations, our task with .egard to it ends with the 
recognition and unequivocal designation of this fact. 
Beyond the recognition of this fact we shall not get, 
and every venture beyond it will only be productive of 
obscurity. All uneasiness will vanish when once we 
nave made clear to ourselves that in the concept of 
mass no theory of any kind whatever is contained, but 
simply a fact of experience. The concept has hitherto 
held good. It is very improbable, but not impossible, 
that it will be shaken in the future, just as the concep- 



222 THE SCIENCE OF MECHANICS. 

tion of a constant quantity of heat, which also rested 
on experience, was modified by new experiences. 
fall XlV 0-f SufJP- to, 3j 

I VI. 

newton's views of time, space, and motion. 

i. In a scholium which he appends immediately to 
his definitions, Newton presents his views regarding 
time and space views which we shall now proceed to 
examine more in detail. We shall literally cite, to this 
end, only the passages that are absolutely necessary 
to the characterisation of Newton's views. 
Newton's " So far, my object has been to explain the senses 

views of . 

time, space, " in which certain words little known are to be used in 

and motion. 

"the sequel. Time, space, place, and motion, being 
" words well known to everybody, I do not define. Yet 
"it is to be remarked, that the vulgar conceive these 
"quantities only in their relation to sensible objects. 
"And hence certain prejudices with respect to them 
"have arisen, to remove which it will be convenient to 
"distinguish them into absolute and relative, true and 
"apparent, mathematical and common, respectively. 
Absolute "I. Absolute, true, and mathematical time, of it- 

and relative ... 

time, " self, and by its own nature, flows uniformly on, witn- 

" out regard to anything external. It is also called 
1 ' duration. 

"Relative, apparent, and common time, is some 
"sensible and external measure of absolute time (dura- 
tion), estimated by the motions of bodies, whether 
"accurate or inequable, and is commonly employed 
" in place of true time ; as an hour, a day, a month, 
" a year. . . 

"The natural days, which, commonly, for the pur- 
" pose of the measurement of time, are held as equal. 
" are in reality unequal. Astronomers correct this in- 



THE PRIXCIPLES OF D YNAMICS. 223 

"equality, in order that they may measure by a truer 
"time the celestial motions. It may be that there is 
"no equable motion, by which time can accurately be 
"measured. All motions can be accelerated and re- 
garded. But the flow of absolute time cannot be 
"changed. Duration, or the persistent existence of 
"things, is always the same, whether motions be swift 
"or slow or null." 

2. It would appear as though Newton in the re- Discussion 

. n e of Newton's 

marks here cited still stood under the influence of the view of 

time. 

mediaeval philosophy, as though he had grown unfaith- 
ful to his resolve to investigate only actual facts. When 
we say a thing A changes with the time, we mean sim- 
ply that the conditions that determine a thing A depend 
on the conditions that determine another thing B. The 
vibrations of a pendulum take place in time when its 
excursion depends on the position of the earth. Since, 
however, in the observation of the pendulum, we are 
not under the necessity of taking into account its de- 
pendence on the position of the earth, but may com- 
pare it with any other thing (the conditions of which 
of course also depend on the position of the earth), the 
illusory notion easily arises that all the things with 
which we compare it are unessential. Nay, we may, 
in attending to the motion of a pendulum, neglect en- 
tirely other external things, and find that for every po- 
sition of it our thoughts and sensations are different. 
Time, accordingly, appears to be some particular and 
independent thing, on the progress of which the posi- 
tion of the pendulum depends, while the things that 
we resort to for comparison and choose at random ap- 
pear to play a wholly collateral part. But we must 
not forget that all things in the world are connected 
with one another and depend on one another, and that 



224 THE SCIENCE OF MECHANICS. 

General we ourselves and all our thoughts are also a part of 

discussion T . . , , , 

of the con- nature. It is utterly beyond our power to measure the 

cept of i - - . - -^ . . . 

time. changes of things by time, (juite the contrary, time 

is an abstraction, at which we arrive by means of the 
changes of things ; made because we are not restricted 
to any one definite measure, all being interconnected. 
A motion is termed uniform in which equal increments 
of space described correspond to equal increments of 
space described by some motion with which we form a 
comparison, as the rotation of the earth. A motion 
may, with respect to another motion, be uniform. But 
the question whether a motion is in itself uniform, is 
senseless. With just as little justice, also, may we 
speak of an "absolute time" of a time independent of 
change. This absolute time can be measured by com- 
parison with no motion ; it has therefore neither a 
practical nor a scientific value ; and no one is justified 
in saying that he knows aught about it. It is an idle 
metaphysical conception. 
Furthereiu- It would not be difficult to show from the points of 
the idea, view of psychology, history, and the science of lan- 
guage (by the names of the chronological divisions), 
that we reach our ideas of time in and through the in- 
terdependence of things on one another. In these ideas 
the profoundest and most universal connection of things j 
is expressed. When a motion takes place in time, it | 
depends on the motion of the earth. This is not refuted 
by the fact that mechanical motions can be reversed. 
A number of variable quantities may be so related that 
one set can suffer a change without the others being 
affected by it. Nature behaves like a machine. The 
individual parts reciprocally determine one another. ! 
But while in a machine the position of one part de- 
termines the position of all the other parts, in nature ' 



THE PRINCIPLES OF DYNAMICS. 225 

more complicated relations obtain. These relations are 
best represented under the conception of a number, 
/;, of quantities that satisfy a lesser number, ri, of equa- 
tions. Were n = ri ', nature would be invariable. Were 
ri = n 1, then with one quantity all the rest would 
be controlled. If this latter relation obtained in na- 
ture, time could be reversed the moment this had been 
accomplished with any one single motion. But the 
true state of things is represented by a different rela- 
tion between n and //'. The quantities in question are 
partially determined by one another ; but they retain 
a greater indeterminateness, or freedom, than in the 
case last cited. We ourselves feel that we are such a 
partially determined, partially undetermined element 
of nature. In so far as a portion only of the changes 
of nature depends on us and can be reversed by us, 
does time appear to us irreversible, and the time that 
is past as irrevocably gone. 

We arrive at the idea of time, to express it briefly Some psy- 
and popularly, by the connection of that which isconsidera- 
contained in the province of our memory with that 
which is contained in the province of our sense-percep- 
tion. When we say that time flows on in a definite di- 
rection or sense, we mean that physical events gene- 
rally (and therefore also physiological events) take 
place only in a definite sense.* Differences of tem- 
perature, electrical differences, differences of level gen- 
erally, if left to themselves, all grow less and not 
greater. If we contemplate two bodies of different 
temperatures, put in contact and left wholly to them- 
selves, we shall find that it is possible only for greater 
differences of temperature in the field of memory to 

* Investigations concerning the physiological nature of the sensations of 
Ume and space are here excluded from consideration. 



T*1c XV -' Su(i|. tt. 2J 



226 



THE SCIENCE OF MECHANICS. 



Newton's 
views of 
space and 
motion. 



Passages 
from his 
works. 



exist with lesser ones in the field of sense-perception, 
and not the reverse. In all this there is simply ex- 
pressed a peculiar and profound connection of things. 
To demand at the present time a full elucidation of this 
matter, is to anticipate, in the manner of speculative 
philosophy, the results of all future special investiga- 
tion, that is a perfect physical science. (Compare Ap- 
pendix, XIX., p. 541.) 

3. Views similar to those concerning time, are de- 
veloped by Newton with respect to space and motion. 
We extract here a few passages which characterise his 
position. 

"II. Absolute space, in its own nature and with- 
"out regard to anything external, always remains sim- 
"ilar and immovable. 

"Relative space is some movable dimension or 
"measure of absolute space, which our senses deter- 
"mine by its position with respect to other bodies,. 
" and which is commonly taken for immovable [abso- \ 
"lute] space. . . . 

" IV. Absolute motion is the translation of a body| 
"from one absolute place* to another absolute place ;j 
" and relative motion, the translation from one relative 
" place to another relative place. . . . 

" . . . . And thus we use, in common affairs, instead j 
"of absolute places and motions, relative ones; anc 
"that without any inconvenience. But in physica 
"disquisitions, we should abstract from the senses 
" For it may be that there is no body really at rest, tc 
"which the places and motions of others can be re 
"f erred. . . . 

" The effects by which absolute and relative motion 



* The place, or locus of a body, according to Newton, is not its positiot j 
but the part of space which it occupies. It is either absolute or relative. Tram 



THE PRIXCIPLES OF DYXAMICS. 1*7 

1 are distinguished from one another, are centrifugal 
' forces, or those forces in circular motion which pro- 
1 duce a tendency of recession from the axis. For in 
a circular motion which is purely relative no such 
'forces exist; but in a true and absolute circular mo- 
'tion they do exist, and are greater or less according 
' to the quantity of the [absolute] motion. 

"For instance. If a bucket, suspended by a long The rota- / 

r j tin 8 bucket/ 

' cord, is so often turned about that finally the cord is 
1 strongly twisted, then is filled with water, and held 
' at rest together with the water ; and afterwards by 
' the action of a second force, it is suddenly set whirl- 
' ing about the contrary way, and continues, while the 
'cord is untwisting itself, for some time in this mo- 
' tion ; the surface of the water will at first be level, 
'just as it was before the vessel began to move ; but, 
' subsequently, the vessel, by gradually communicat- 
' ing its motion to the water, will make it begin sens- 
' ibly to rotate, and the water will recede little by little 
' from the middle and rise up at the sides of the ves- 
' sel, its surface assuming a concave form. (This ex- 
periment I have made myself.) 

" .... At first, when the relative motion of the wa- Relative 

and real 

' ter in the vessel was greatest, that motion produced motion. 
' no tendency whatever of recession from the axis ; the 
' water made no endeavor to move towards the cir- 
' cumference, by rising at the sides of the vessel, but 
' remained level, and for that reason its true circular 
' motion had not yet begun. But afterwards, when 
' the relative motion of the water had decreased, the 
' rising of the water at the sides of the vessel indicated 
' an endeavor to recede from the axis ; and this en- 
' deavcr revealed the real circular motion of the water, 
' continually increasing, till it had reached its greatest 



228 



THE SCIENCE OF MECHANICS. 



Newton's 
criteria for 
distinguish- 
ing absolute 
from rela- 
tive motion. 



point, when relatively the water was at rest in the 
vessel .... 

" It is indeed a matter of great difficulty to discover 
and effectually to distinguish the true from the ap- 
parent motions of particular bodies ; for the parts of 
that immovable space in which bodies actually move, 
do not come under the observation of our senses. 

" Yet the case is not altogether desperate ; for there 
exist to guide us certain marks, abstracted partly 
from the apparent motions, which are the differences 
of the true motions, and partly from the forces that 
are the causes and effects of the true motions. If, 
for instance, two globes, kept at a fixed distance 
from one another by means of a cord that connects 
them, be revolved about their common centre of 
gravity, one might, from the simple tension of the 
cord, discover the tendency of the globes to recede 
from the axis of their motion, and on this basis the 
quantity of their circular motion might be computed. 
And if any equal forces should be simultaneously 
impressed on alternate faces of the globes to augment 
or diminish their circular motion, we might, from 
the increase or decrease of the tension of the cord, 
deduce the increment or decrement of their motion; 
and it might also be found thence on what faces 
forces would have to be impressed, in order that the 
motion of the globes should be most augmented ; 
that is, their sear faces, or those which, in the cir- 
cular motion, follow. But as soon as we knew which 
faces followed, and consequently which preceded, we 
should likewise know the direction of the motion. 
In this way we might find both the quantity and the 
: direction of the circular motion, considered even in 
: an immense vacuum, where there was nothing ex- 



THE PRIXCIPLES OF D YNAMICS. 229 

"ternal or sensible with which the globes could be 
"compared "Add *V// rf $ff' [.ft 

4. It is scarcely necessary to remark that in the re- Thepredi- 

i-i cations of 

flections here presented Newton has again acted con- Newton 

. . .are not the 

trary to his expressed intention only to investigate actual expression 

v > 1 l of actual 

facts, No one is competent to predicate things about facts, 
absolute space and absolute motion; they are pure 
things of thought, pure mental constructs, that cannot 
be produced in experience. All our principles of me- 
chanics are, as we have shown in detail, experimental 
knowledge concerning the relative positions and mo- 
tions of bodies. Even in the provinces in which they 
are now recognised as valid, they could not, and were 
not, admitted without previously being subjected to 
experimental tests. No one is warranted in extending 
these principles beyond the boundaries of experience. 
In fact, such an extension is meaningless, as no one 
possesses the requisite knowledge to make use of it. 






Let us look at the matter in detail. When we say that Detailed 

. view of the 

a body K alters its direction and velocity solely through matter, 
the influence of another body A", we have asserted 
a conception that it is impossible to come at unless 
other bodies A, B, C . . . . are present with reference 
to which the motion of the body K has been estimated. 
In reality, therefore, we are simply cognisant of a re- 
lation of the body K to A, B, C . . . . If now we sud- 
denly neglect A, B, C. . . . and attempt to speak of 
the deportment of the body K in absolute space, we 
implicate ourselves in a twofold error. In the first 
place, we cannot know how K would act in the ab- 
sence of A, B, C . ... ; and in the second place, every 
means would J^e wanting of forming a judgment of the 
behaviour of K and of putting to the test what we had 



230 THE SCIENCE OF MECHANICS. 

predicated,- which latter therefore would be bereft of 
all scientific significance. 
The part Two bodies K and K ', which gravitate toward each 

which the ... ... 

bodies of other, impart to each other in the direction of their 

space play ... ... . . . 

inthede- line of junction accelerations inversely proportional to 

termination . . ... . 

of motion, their masses in, m . In this proposition is contained, 
not only a relation of the bodies K and K' to one an- 
other, but also a relation of them to other bodies. For 
the proposition asserts, not only that K and K' suffer 
with respect to one another the acceleration designated 
by k (in -f- m'/r 2 ), but also that K experiences the ac- 
celeration Km /r 2 and K' the acceleration -f- Km/r 2 
in the direction of the line of junction ; facts which can 
be ascertained only by the presence of other bodies. 

The motion of a body K can only be estimated by 
reference to other bodies A, B, C . . . . But since we 
always have at our disposal a sufficient number of 
bodies, that are as respects each other relatively fixed, 
or only slowly change their positions, we are, in such 
reference, restricted to no one definite body and can 
alternately leave out of account now this one and now 
that one. In this way the conviction arose that these 
bodies are indifferent generally. 

Thehy- It might be, indeed, that the isolated bodies A, B, 

pothesis of 

a medium C . . . . play merely a collateral role in the determina- 

in space de- 
terminative tion of the motion of the body K, and that this motion 

of motion. . _ 

is determined by a medium in which K exists. In such 
a case we should have to substitute this medium for 
Newton's absolute space. Newton certainly did not 
entertain this idea. Moreover, it is easily demonstrable 
that the atmosphere is not this motion-determinative 
medium. We should, therefore, have, to picture to 
ourselves some other medium, filling, say, all space, 
with respect to the constitution of which and its kinetic 



THE PRIXCIPLES OF DYNAMICS. 231 

relations to the bodies placed in it we have at present 
no adequate knowledge. In itself such a state of things 
would not belong to the impossibilities. It is known, 
from recent hydrodynamical investigations, that a rigid 
body experiences resistance in a frictionless fluid only 
when its velocity changes. True, this result is derived 
theoretically from the notion of inertia ; but it might, 
conversely, also be regarded as the primitive fact from 
which we have to start. Although, practically, and at 
present, nothing is to be accomplished with this con- 
ception, we might still hope to learn more in the future 
concerning this hypothetical medium ; and from the 
point of view of science it would be in every respect 
a more valuable acquisition than the forlorn idea^of_ 
absolute space. When we reflect that we cannot abol- 
ish the isolated bodies A, B, C . . . ., that is, cannot 
determine by experiment whether the part they play is 
fundamental or collateral, that hitherto they have been 
the sole and only competent means of the orientation 
of motions and of the description of mechanical facts, 
it will be found expedient provisionally to regard all 
motions as determined by these bodies. 
1/ 5. Let us now examine the point on which New- Critical 

. examina- 

ton, apparentlv with sound reasons, rests his distinc- tion of 

rr J Newton's 

tion of absolute and relative motion. If the earth is distinction 

of absolute 

affected with an absolute rotation about its axis, cen- from rela- 
tive motion. 

tnfugal forces are set up in the earth : it assumes an 
oblate form, the acceleration of gravity is diminished 
at the equator, the plane of Foucault's pendulum ro- 
tates, and so on. All these phenomena disappear if 
the earth is at rest and the other heavenly bodies are 
affected with absolute motion round it, such that the 
same relative rotation is produced. This is, indeed, the 
case, if we start ab initio from the idea of absolute space. 



232 THE SCIENCE OF MECHANICS. 

But if we take our stand on the basis of facts, we shall 
find we have knowledge only of relative spaces and mo- 
tions. Relatively, not considering the unknown and 
neglected medium of space, the motions of the uni- 
verse are the same whether we adopt the Ptolemaic or 
the Copernican mode of view. Both views are, indeed, 
equally correct ; only the latter is more simple and more 
practical. The universe is not twice given, with an 
earth at rest and an earth in motion ; but only once, 
with its relative motions, alone determinable. It is, 
accordingly, not permitted us to say how things would 
be if the earth did not rotate. We may interpret the 
one case that is given us, in different ways. If, how- 
ever, we so interpret it that we come into conflict with 
experience, our interpretation is simply wrong. The 
principles of mechanics can, indeed, be so conceived, 
that even for relative rotations centrifugal forces arise, 
interpreta- Newton's experiment with the rotating vessel of 

tion of the . ... 

experiment water simply informs us, that the relative rotation of 

with the 

rotating the water with respect to the sides of the vessel pro- 
bucket of . 
water. duces no noticeable centrifugal forces, but that such 

forces are produced by its relative rotation with respect 

to the mass of the earth and the other celestial bodies. 

No one is competent to say how the experiment would ; 

turn out if the sides of the vessel increased in thickness j 

and mass till they were ultimately several leagues thick. ! 

The one experiment only lies before us, and our busi- 

ness is, to bring it into accord with the other facts | 

known to us, and not with the arbitrary fictions of our j 

imagination. \\ 

, vivif 6- We can have no doubts concerning the signifi- 

t cance of the law of inertia if we bear in mind the man- \ 

\ ner in which it was reached. To begin with, Galileo 

\ discovered the constancy of the velocity and direction j 



THE PRINCIPLES OF DYNAMICS. 233 

of a bodv referred to terrestrial objects. Most terres- The law of 

J . inertia in 

trial motions are of such brief duration and extent, that the light of 

this view. 

it is wholly unnecessary to take into account the earth's 
rotation and the changes of its progressive velocity with 
respect to the celestial bodies. This consideration is 
found necessary only in the case of projectiles cast 
great distances, in the case of the vibrations of Fou- 
cault's pendulum, and in similar instances. When now 
Newton sought to apply the mechanical principles dis- 
covered since Galileo's time to the planetary system, 
he found that, so far as it is possible to form any es- 
timate at all thereof, the planets, irrespectively of dy- 
namic effects, appear to preserve their direction and 
velocity with respect to bodies of the universe that are 
very remote and as regards each other apparently fixed, 
. the same as bodies moving on the earth do with re- 
[spect to the fixed objects of the earth. The comport- 
ment of terrestrial bodies with respect to the earth is 
reducible to the comportment of the earth with respect 
to the remote heavenly bodies. If we were to assert 
that we knew more of moving objects than this their 
last -mentioned, experimentally -given comportment 
with respect to the celestial bodies, we should render 
ourselves culpable of a falsity. When, accordingly, we 
say, that a body preserves unchanged its direction and 
velocity in space, our assertion is nothing more or less 
than an abbreviated reference to the entire universe. 
The use of such an abbreviated expression is permit- 
ted the original author of the principle, because he 
knows, that as things are no difficulties stand in the 
way of carrying out its implied directions. But no 
remedy lies in his power, if difficulties of the kind men- 
tioned present themselves ; if, for example, the re- 
quisite, relatively fixed bodies are wanting. 



234 . THE SCIENCE OF MECHANICS. 

Thereia- 7. Instead, now, of referring a moving body K to 

tion of the . . 

bodies of space, that is to say to a s\'stem of coordinates, let us 

the uni- . 

verse to view directly its relation to the bodies of the universe, 

each other. . ' 

by which alone such a system of coordinates can be 
determined. Bodies very remote from each other, mov- 
ing with constant direction and velocity with respect 
to other distant fixed bodies, change their mutual dis- 
tances proportionately to the time. We may also say, 
All very remote bodies all mutual or other forces ne- 
glected alter their mutual distances proportionately 
to those distances. Two bodies, which, situated at a 
short distance from one another, move with constant 
direction and velocity with respect to other fixed bod- 
ies, exhibit more complicated relations. If we should 
regard the two bodies as dependent on one another, 
and call r the distance, / the time, and a a constant 
dependent on the directions and velocities, the formula 
would be obtained: d 2 r/dt 2 = (1/r) [a 2 (dr/dl) 2 }. 
It is manifestly much simpler and clearer to regard the 
two bodies as independent of each other and to con- 
sider the constancy of their direction and velocity with 
respect to other bodies. 

Instead of saying, the direction and velocity of a 
mass fA. in space remain constant, we may also employ 
the expression, the mean acceleration of the mass // 
with respect to the masses ;;/, m', m" . ... at the dis- 
tances r, r', r". ... is = 0, or d 2 {2 m rj 2 m) jd t 2 0. 
The latter expression is equivalent to the former, as 
soon as we take into consideration a sufficient number j 
of sufficiently distant and sufficiently large masses. j 
The mutual influence of more proximate small masses, 
which are apparently not concerned about each other, 
is eliminated of itself. That the constancy of direction 
and velocity is given by the condition adduced, will be 



THE PRIXCIPLES OF DYNAMICS. 235 

seen at once if we construct through // as vertex cones The expres- 

sion of the 

that cut out different portions of space, and set up the law of iner- 

. . tia in terms 

condition with respect to the masses of these separate of this re- 

, . lation. 

portions. We may put, indeed, for the entire space 
encompassing /v, d 2 {2 mr/2 ni) /dt 2 = 0. But the 
equation in this case asserts nothing with respect to the 
motion of /u, since it holds good for all species of mo- 
tion where // is uniformly surrounded by an infinite 
number of masses. If two masses pi 1} pi 2 exert on each 
other a force which is dependent on their distance r, 
then d 2 r/dt 2 = O x + /i 2 )/(r). But, at the same time, 
the acceleration of the centre of gravity of the two 
masses or the mean acceleration of the mass-system 
with respect to the masses of the universe (by the prin- 
ciple of reaction) remains = ; that is to say, 



d 2 

IT* 



2mr. , 2mr, 

Pt-TTzr + H 



2 m ' r2 2 m 



= 0. 



When we reflect that the time-factor that enters The neces- 
sity in sci- 
mto the acceleration is nothing more than a quantity ence of a 

. considera- 

that is the measure of the distances (or angles of rota- tion of the 

, . ah. 

tion) of the bodies of the universe, we see that even in 

the simplest case, in which apparently we deal with 
the mutual action of only tzuo masses, the neglecting 
of the rest of the \\ odd is impossible. Nature does not 
begin with elements, as we are obliged to begin with 
them. It is certainly fortunate for us, that we can, 
from time to time, turn aside our eyes from the over- 
powering unity of the All, and allow them to rest on 
individual details. But we should not omit, ultimately 
to complete and correct our views by a thorough con- 
sideration of the things which for the time being we 
left out of account. 

8. The considerations just presented show, that it 



236 



THE SCIENCE OF MECHANICS. 



The law of 
inertia does 
not involve 
absolute 
space. 



Natural 
processes 
consist in 
the equali- 
sation of 
the differ- 
ences of 
quantities. 



is not necessary to refer the law of inertia to a special 
absolute space. On the contrary, it is perceived that 
the masses that in the common phraseology exert forces 
on each other as well as those that exert none, stand 
with respect to acceleration in quite similar relations. 
We may, indeed, regard all masses as related to each 
other. That accelerations play a prominent part in the 
relations of the masses, must be accepted as a fact of 
experience ; which does not, however, exclude attempts 
to elucidate this fact by a comparison of it with other 
facts, involving the discovery of new points of view. 
In all the processes of nature the differences of certain 

quantities u play a de- 
terminative role. Differ- 
ences of temperature, of 
potential function, and so 
forth, induce the natural 
processes, which consist 
in the equalisation of 




-*P 



Fig. 143. 

these differences. The familiar expressions d 2 u/dx i f 
d 2 u/dj 2 , d 2 u/dz 2 , which are determinative of the 
character of the equalisation, may be regarded as the 
measure of the departure of the condition of any point 
from the mean of the conditions of its environment 
to which mean the point tends. The accelerations of 
masses may be analogously conceived. The great dis- 
tances between masses that stand in no especial force- 
relation to one another, change proportionately to each 
other. If we lay off, therefore, a certain distance p as 
abscissa, and another r as ordinate, we obtain a straight 
line. (Fig. 143.) Every r-ordinate corresponding to 
a definite p-value represents, accordingly, the mean of 
the adjacent ordinates. If a force-relation exists be- 
tween the bodies, some value d 2 r/dt 2 is determined 



THE PRINCIPLES OF DYNAMICS. 237 

by it which conformably to the remarks above we may 
replace by an expression of the form d 2 r/dp 2 . By the 
force-relation, therefore, a departure of the r-ordinate 
from the mean of the adjacent ordinates is produced, 
which would not exist if the supposed force-relation 
did not obtain. This intimation will suffice here. 

g. We have attempted in the foregoing to give the character 

, . of the new 

law of inertia a different expression from that in ordi- expression 

rr for the law 

nary use. This expression will, so long as a surh- of inertia. 

cient number of bodies are apparently fixed in space, 

accomplish the same as the ordinary one. It is as 

easily applied, and it encounters the same difficulties. 

In the one case we are unable to come at an absolute 

space, in the other a limited number of masses only is 

within the reach of our knowledge, and the summation 

indicated can consequently not be fully carried out. It 

is impossible to say whether the new expression would 

still represent the true condition of things if the stars 

were to perform rapid movements among one another. 

The general experience cannot be constructed from the 

particular case given us. We must, on the contrary, 

wait until such an experience presents itself. Perhaps 

when our physico-astronomical knowledge has been 

extended, it will be offered somewhere in celestial 

space, where more violent and complicated motions 

take place than in our environment. The most impor- The sim- 
plest prin- 
tant result of our reflexions is, however, that precisely cipiesof 

mechanics 

the apparently simplest mechanical principles are of a very are of a 

rl ' 2 . f r j j highly com- 

compluated character, that these principles are founded on plicated na- 
ture and are 
uncompleted experiences, nay on experiences that never can ail derived 

from expe- 

oe fully completed, that practically, indeed, they are suf rience. 
ficiently secured, in view of the tolerable stability of our 
environment, to serve as the foundation of mathematical 
deduction, but that they can by no means themselves be re- 



233 THE SCIENCE OF MECHANICS. 

garded as mathematically established truths but only a: 
principles that not only admit of constant control by expe 
rience but actually require it. f This perception is valu 
able in that it is propitious to the advancement o 
science. (Compare Appendix, XX., p. 542.) 



SYNOPTICAL CRITIQUE OF THE NEWTONIAN ENUNCIATIONS < 

Newton's i. Now that we have discussed the details witl; 

Definitions. _ . ... 

sufficient particularity, we may pass again under re j 
view the form and the disposition of the Newtonian 
enunciations. Newton premises to his work several 
definitions, following which he gives the laws of mci 
tion. We shall take up the former first. 

Mass. " Definition I. The quantity of any matter is thj 

"measure of it by its density and volume conjointly 
"... This quantity is what I shall understand by th 
" term mass or body in the discussions to follow. It i 
" ascertainable from the weight of the body in que; 
"tion. For I have found, by pendulum-experiment 
"of high precision, that the mass of a body is propo;: 
" tional to its weight ; as will hereafter be shown. 

Quantity of "Definition II. Quantity of motion is the measur, 

motion, . J 

inertia, " of it by the velocity and quantity of matter cor; 

force, and . . J J ^ J 

accelera- "jointly. 

" Definition III. The resident force [vis insita, i. < 
"the inertia] of matter is a power of resisting, b| 
"which every body, so far as in it lies, perseveres i 
"its state of rest or of uniform motion in a straigl; 
"line. 

"Definition IV. An impressed force is any actio 
"upon a body which changes, or tends to change, i 
"state of rest, or of uniform motion in a straight lin< 



tion. 



THE PRIXCIPLES OF DYNAMICS. 239 

"Definition V. A centripetal force is any force by 
"which bodies are drawn or impelled towards, or tend 
"in any way to reach, some point as centre. 

"Definition VI. The absolute quantity of a centri- Forcescias- 

... . .... sifted as ab- 

" petal force is a measure of it increasing and dimin- solute, ac- 

... _. . , , celerati%-e, 

"lshmg with the efficacy of the cause that propagates and mov- 

ing. 

"it from the centre through the space round about. 

"Definition VII. The accelerative quantity of a 
"centripetal force is the measure of it proportional to 
" the velocity which it generates in a given time. 

"Definition VIII The moving quantity of a cen- 
" tripetal force is the measure of it proportional to the 
"motion [See Def. 11.] which it generates in a given 
" time. 

"The three quantities or measures of force thus dis- The reia- 

i_ tions o the 

"tinguished, may, for brevity's sake, be called abso- forces thus 

. distin- 

"lute, accelerative, and moving forces, being, for dis-guished. 
"tinction's sake, respectively referred to the centre of 
"force, to the places of the bodies, and to the bodies 
' ' that tend to the centre : that is to say, I refer moving 
"force to the body, as being an endeavor of the whole 
towards the centre, arising from the collective en- 
"deavors of the several parts ; accelerative force to the 
"place of the body, as being a sort of efficacy originat- 
" ing in the centre and diffused throughout all the sev- 
"eral places round about, in moving the bodies that 
" are at these places ; and absolute force to the centre, 
"as invested with some cause, without which moving 
"forces would not be propagated through the space 
" round about ; whether this latter cause be some cen- 
" tral body, (such as is a loadstone in a centre of mag- 
netic force, or the earth in the centre of the force of 
"gravity,) or anything else not visible. This, at least, 
"is the mathematical conception of forces; for their 



240 



THE SCIENCE OF MECHANICS. 



The dis- 
tinction 
mathemat- 
ical and not 
physical. 



' physical causes and seats I do not in this place con- 
' sider. 

"Accelerating force, therefore, is to moving force, 
' as velocity is to quantity of motion. For quantity 
' of motion arises from the velocity and the quantity 
' of matter ; and moving force arises from the accel- 
' erating force and the same quantity of matter ; the 
' sum of the effects of the accelerative force on the sev- 
' eral particles of the body being the motive force of 
' the whole. Hence, near the surface of the earth, 
f where the accelerative gravity or gravitating force is 
'in all bodies the same, the motive force of gravity 01 
' the weight is as the body [mass]. But if we ascend ] 
'to higher regions, where the accelerative force oil 
' gravity is less, the weight will be equally diminished. 1 
' always remaining proportional conjointly to the mas'j 
' and the accelerative force of gravity. Thus, in those! 
' regions where the accelerative force of gravity is hal i 
' as great, the weight of a body will be diminished b}; 
'one- half. Further, I apply the terms accelerative anc 
'motive in one and the same sense to attractions anc! 
' to impulses. I employ the expressions attraction, im | 
'pulse, or propensity of any kind towards a centre 
' promiscuously and indifferently, the one for the other; 
' considering those forces not in a physical sense, buj 
'mathematically. The reader, therefore, must noj 
'infer from any expressions of this kind that I ma 
' use, that I take upon me to explain the kind or th ' 
' mode of an action, or the causes or the physical reaj 
'son thereof, or that I attribute forces in a true c 
'physical sense, to centres (which are only mathemat 
' ical points), when at any time I happen to say tha! 
' centres attract or that central forces are in action. 



THE PRIXCIPLES OF DYNAMICS. 241 

2. Definition 1 is, as has already been set forth, acmicismof 

Newton's 

pseudo-definition. The concept of mass is not made Definitions. 
clearer by describing mass as the product of the volume 
into the density, as density itself denotes simply the 
mass of unit of volume. The true definition of mass 
can be deduced only from the dynamical relations of 
bodies. 

To Definition 11, which simply enunciates a mode 
of computation, no objection is to be made. Defini- 
tion in (inertia), however, is rendered superfluous by 
Definitions iv-vin of force, inertia being included and 
given in the fact that forces are accelerative. 

Definition iv defines force as the cause of the accel- 
eration, or tendency to acceleration, of a body. The 
latter part of this is justified by the fact that in the 
cases also in which accelerations cannot take place, 
other attractions that answer thereto, as the compres- 
sion and distension etc. of bodies occur. The cause 
of an acceleration towards a definite centre is defined 
in Definition v as centripetal force, and is distinguished 
in vi, vii, and vm as absolute, accelerative, and mo- 
tive. It is, we may say, a matter of taste and of form 
whether we shall embody the explication of the idea 
of force in one or in several definitions. In point of 
principle the Newtonian definitions are open to no ob- 
jections. 

3. The Axioms or Laws of Motion then follow, of Newton's 

Laws of 

which Newton enunciates three : Motion. 

" Law I. Every body perseveres in its state of rest 
"or of uniform motion in a straight line, except in so 
"far as it is compelled to change that state by im- 
" pressed forces." 

"Law LL. Change of motion [i. e. of momentum] is 
' proportional to the moving force impressed, and takes 



Il 1 



242 7Y/ SCIENCE OF MECHANICS. 

"place in the direction of the straight line in which 
"such force is impressed. " 

11 Law III. Reaction is always equal and opposite 
"to action; that is to say, the actions of two bodies 
"upon each other are always equal and directly op- 
" posite." 

Newton appends to these three laws a number of 
Corollaries. The first and second relate to the prin- 
ciple of the parallelogram of forces ; the third to the 
quantity of motion generated in the mutual action of 
bodies ; the fourth to the fact that the motion of the 
centre of gravity is not changed by the mutual action 
of bodies ; the fifth and sixth to relative motion. 
Criticism of 4. We readily perceive that Laws i and n are con- 

Newton's 

laws of tained in the definitions of force that precede. Ac- 
cording to the latter, without force there is no accel- 
eration, consequently only rest or uniform motion in a 
straight line. Furthermore, it is wholly unnecessary 
tautology, after having established acceleration as the 
measure of force, to say again that change of motion is 
proportional to the force. It would have been enougl \ 
to say that the definitions premised were not arbitrary 
mathematical ones, but correspond to properties o ' 
bodies experimentally given. The third law apparently 
contains something new. But we have seen that it i 
unintelligible without the correct idea of mass, whicl 1 
idea, being itself obtained only from dynamical expe 
rience, renders the law unnecessary. 



motion. 



Thecoroi- The first corollary really does contain somethin 

laries to . . . . 

these laws. new. But it regards the accelerations determined I 
a body K by different bodies M, N, P as self-evidentt 
independent of each other, whereas this is precise!; 
what should have been explicitly recognised as a /at 
of experience. Corollary Second is a simple applies I 



THE PRIXCIPLES OF DYNAMICS. 243 

tion of the law enunciated in corollary First. The re- 
maining corollaries, likewise, are simple deductions, 
that is, mathematical consequences, from the concep- 
tions and laws that precede. 

5. Even if we adhere absolutely to the Newtonian 
points of view, and disregard the complications and in- 
definite features mentioned, which are not removed 
but merely concealed by the abbreviated designations 
Time" and "Space," it is possible to replace New- 
ton's enunciations by much more simple, methodically 
better arranged, and more satisfactory propositions. 
Such, in our estimation, would be the following : 

a. Experimental Proposition. Bodies set opposite Proposed 

* . . substitu- 

each other induce in each other, under certain circum- tions for 

the New- 
stances to be specified by experimental physics, con- tonian laws 

- . an d denni- 

trary accelerations in the direction of their line of junc- tions. 
tion. (The principle of inertia is included in this.) 

b. Definition. The mass-ratio of any two bodies is 
the negative inverse ratio of the mutually induced ac- 
celerations of those bodies. 

c. Experimental Proposition. The mass-ratios of 
bodies are independent of the character of the physical 
states (of the bodies) that condition the mutual accel- 
erations produced, be those states electrical, magnetic, 
or what not ; and they remain, moreover, the same, 
whether they are mediately or immediately arrived at. 

d. Experimental Proposition. The accelerations 
which any number of bodies A, B, C . . . . induce in a 
body K, are independent of each other. (The principle 
of the parallelogram of forces follows immediately from 
this.) 

e. Definition. Moving force is the product of the 
mass-value of a body into the acceleration induced in 
that body. 4^ V)(; ,| u. />.-4(, 



244 THE SCIENCE OF MECHANICS. 

Extent and Then the remaining arbitrary definitions of the al- 

character , . 

of the pro- gebraical expressions "momentum," "vis viva," and 

posed sub- * 

stitutions. the like, might follow. But these are by no means in- 
dispensable. The propositions above set forth satisfy 
the requirements of simplicity and parsimony which, 
on economico-scientific grounds, must be exacted of 
them. They are, moreover, obvious and clear ; for no 
doubt can exist with respect to any one of them either 
concerning its meaning or its source ; and we always 
know whether it asserts an experience or an arbitrary 
convention. 

The 6. Upon the whole, we may say, that Newton dis 

achieve- . 

ments of cerned in an admirable manner the concepts and princi 

Newton 

from the pies that were sufncie?itly assured to allow of being fur 

point of \ ........ & 

view of his ther built upon. It is possible that to some extent In 

time. 

was forced by the difficulty and novelty of his subject j 
in the minds of the contemporary world, to great am 
plitude, and, therefore, to a certain disconnectednes I 
of presentation, in consequence of which one and th< 
same property of mechanical processes appears severa 
times formulated. To some extent, however, he was! 
as it is possible to prove, not perfectly clear himsel| 
concerning the import and especially concerning th; 
source of his principles. This cannot, however, ol 
scure in the slightest his intellectual greatness. 1 1 
that has to acquire a new point of view naturally cai 
not possess it so securely from the beginning as the 
that receive it unlaboriously from him. He has don 
enough if he has discovered truths on which futm 
generations can further build. For every new infe 
ence therefrom affords at once a new insight, a nc 
control, an extension of our prospect, and a clarific; 
tion of our field of view. Like the commander of a 
army, a great discoverer cannot stop to institute petf 



THE PRINCIPLES OF DYNAMICS. 245 

inquiries regarding the right by which he holds each The 
post of vantage he has won. The magnitude of thememsof 

... Newton in 

problem to be solved leaves no time for this. But at the light of 

. subsequent 

a later period, the case is different. Newton might research, 
well have expected of the two centuries to follow that 
they should further examine and confirm the founda- 
tions of his work, and that, when times of greater scien- 
tific tranquillity should come, the principles of the sub- 
ject might acquire an even higher philosophical in- 
terest than all that is deducible from them. Then prob- 
lems arise like those just treated of, to the solution of 
which, perhaps, a small contribution has here been 
made. We join with the eminent physicists T homso n tor/ n&fy 
and Tait , in our reverence and admiration of Newton. 
But we can only comprehend with difficulty their opin- 
ion that the Newtonian doctrines still remain the best 
and most philosophical foundation of the science that 
can be given. 



RETROSPECT OF THE DEVELOPMENT OF DYNAMICS. 

i. If we pass in review the period in which the de- The chief 

result, the 

velopment of dynamics fell, a period inaugurated by discovery 
Galileo, continued by Huygens, and brought to a close fact. 
by Newton, its main result will be found to be the 
perception, that bodies mutually determine in each 
other accelerations dependent on definite spatial and 
material circumstances, and that there are masses. The 
reason the perception of these facts was embodied in 
so great a number of principles is wholly an historical 
one ; the perception was not reached at once, but slowly 
and by degrees. In reality only one great fact was es- 
tablished. Different pairs of bodies determine, inde- 
pendently of each other, and mutually, in themselves, 



246 THE SCIENCE OF MECHANICS. 

pairs of accelerations, whose terms exhibit a constant 
ratio, the criterion and characteristic of each pair. 
This fact Not even men of the calibre of Galileo, Huygens, 

even the .... 

greatest in- and JNewton were able to perceive this fact at once, 
could per- Even they could only discover it piece by piece, as it 

ccivc only 

in frag- is expressed in the law of falling bodies, in the special 



inents. 



law of inertia, in the principle of the parallelogram of 
forces, in the concept of mass, and so forth. To-day, 
no difficulty any longer exists in apprehending the unity 
of the whole fact. The practical demands of communi- 
cation alone can justify its piecemeal presentation in 
several distinct principles, the number of which is really 
only determined by scientific taste. What is more, a 
reference to the reflections above set forth respecting 
the ideas of time, inertia, and the like, will surely con- 
vince us that, accurately viewed, the entire fact has, 
in all its aspects, not yet been perfectly apprehended. 
The results The point of view reached has, as Newton expressly 
havenoth- states, nothing to do with the " unknown causes " of 
with the so- natural phenomena. That which in the mechanics of 
"causes" the present day is called force is not a something that 

of phenom- . . 

ena. lies latent m the natural processes, but a measurable, 

actual circumstance of motion, the product of the mass 
into the acceleration. Also when we speak of the at- 
tractions or repulsions of bodies, it is not necessary to 
think of any hidden causes of the motions produced. 
We signalise by the term attraction merely an actually 
existing resemblance between events determined by con- 
ditions of motion and the results of our volitional im- 
pulses. In both cases either actual motion occurs or, 
when the motion is counteracted by some other circum- 
stance of motion, distortion, compression of bodies, 
and so forth, are produced. 

2. The work which devolved on genius here, was 



THE PRINCIPLES OF DYNAMICS. 247 

the noting of the connection of certain determinative The form of 

_,. the me- 

elements of the mechanical processes. The precise es- chanicai 

1 principles. 

tablishment of the form of this connection was rather a m the main 

1 1 i-i j i j tt ' histor- 

task for plodding research, which created the different icai origin, 
concepts and principles of mechanics. We can de- 
termine the true value and significance of these prin- 
ciples and concepts only by the investigation of their 
historical origin. In this it appears unmistakable at 
times, that accidental circumstances have given to the 
course of their development a peculiar direction, which 
under other conditions might have been very different. 
Of this an example shall be given. 

Before Galileo assumed the familiar fact of the de-Forexam- 

r 1 J P' e> Gali- 

pendence of the final velocitv on the time, and put it to leos laws 

' of falling 

the test of experiment, he essayed, as we have alrearay bodies 

% r * m 'R nt have 

seen, a different hypothesis, and made the final velocitv taken a dif- 

1 tt j l ' ferentform. 

t ^Unproportional to the space described. He imagined, by a 
jrfcourse of fallacious reasoning, likewise already referred 
to, that this assumption involved a self-contradiction. 
His reasoning was, that twice any given distance of de- 
scent must, by virtue of the double final velocity ac- 
quired, necessarily be traversed in the same time as the 
simple distance of descent. But since the first half is 
necessarily traversed first, the remaining half will have 
to be traversed instantaneously, that is in an interval 
of time not measurable. Whence, it readily follows, 
that the descent of bodies generally is instantaneous. 

The fallacies involved in this reasoning are manifest. Galileo's 

r~. r j - 1 reasoning 

Galileo was, of course, not versed in mental Integra- and its 
tibns, and having at his command no adequate methods 
for the solution of problems whose facts were in any 
degree complicated, he could not but fall into mistakes 
whenever such cases were presented. If we call s the 
distance and / the time, the Galilean assumption reads 



248 



THE SCIENCE OF MECHANICS, 



The suppo- 
sition that 
Kepler had 
made Gali- 
leo's re- 
searches. 




In such a 
case the 
concept 
" work " 
might have 
been the 
original 
concept of 
mechanics. 



in the language of to-day ds/dt as, from which fol- 
lows s = A "', where a is a constant of experience and 
A a constant of integration. This is an entirely different 
conclusion from that drawn by Galileo. It does not 
conform, it is true, to experience, and Galileo would 
probably have taken exception to a result that, as a 
condition of motion generally, made s different from 
when / equalled 0. But in itself the assumption is by 
no means .sr //"-contradictory. 

Let us suppose that Kepler had put to himself the 
same question. Whereas Galileo always sought after 
the very simplest solutions of things, and at once re- 
jected hypotheses that did not fit, Kepler's mode of pro- 
cedure was entirely different. He did not quail before 
the most complicated assumptions, but worked his way, 
by the constant gradual modification of his original 
hypothesis, successfully to his goal, as the history of 
his discovery of the laws of planetary motion fully 
shows. Most likely, Kepler, on finding the assumption 
ds/dt == as would not work, would have tried a num- 
ber of others, and among them probably the correct one 
ds/dt = a \ / s. But from this would have resulted an 
essentially different course of development for the sci- 
ence of dynamics. 

It was only gradually and with great difficulty that 
the concept of "work" attained its present position 
of importance ; and in our judgment it is to the above- 
mentioned trifling historical circumstance that the diffi- 
culties and obstacles it had to encounter are to be as- 
cribed. As the interdependence of the velocity and the 
time was, as it chanced, first ascertained, it could not 
be otherwise than that the relation v = gt should appear 
as the original one, the equation s = gt 2 /2 as the next 
immediate, and gs = v 2 /2 as a remoter inference. In- 



THE PRINCIPLES OF DYNAMICS. 249 

troducing the concepts mass (///) and force (J>), where 
p =r mg, we obtain, by multiplying the three equations 
by m, the expressions mv=pt, ms =pt 2 {2, ps = 
mv*/i the fundamental equations of mechanics. Of 
necessity, therefore, the concepts force and momentum 
(m v ) appear more primitive than the concepts work (J> s) 
and vis viva (mv 2 ). It is not to be wondered at, accord- 
ingly, that, wherever the idea of work made its appear- 
ance, it was always sought to replace it by the histor- 
ically older concepts. The entire dispute of the Leib- 
nitzians and Cartesians, which was first composed in 
a manner by D'Alembert, finds its complete explana- 
tion in this fact. 

From an unbiassed point of view, we have exactly Justifica- 
the same right to inquire after the interdependence of view. 
the final velocity and the time as after the interde- 
pendence of the final velocity and the distance, and to 
answer the question by experiment. The first inquiry 
leads us to the experiential truth, that given bodies in 
contraposition impart to each other in given times defi- 
nite increments of velocity. The second informs us, 
that given bodies in contraposition impart to each other 
for given mutual displacements definite increments of 
velocities. Both propositions are equally justified, and 
both may be regarded as equally original. 

The correctness of this view has been substantiated ExempHfi- 

cation of it 

in our own day by the example of J. R. Mayer. Mayer, in modem 
a modern mind of the Galilean stamp, a mind wholly 
free from the influences of the schools, of his own in- 
dependent accord actually pursued the last-named 
method, and produced by it an extension of science 
which the schools did not accomplish until later in a 
much less complete and less simple form. For Mayer, 
work was the original concept. That which is called 



250 THE SCIENCE OF MECHANICS. 

work in the mechanics of the schools, he calls force. 
Mayer's error was, that he regarded his method as the 
only correct one. 
The results 3. We may, therefore, as it suits us, regard the time 

which flow - , i7- e t 

from it. of descent or the distance of descent as the factor de- 
terminative of velocity. If we fix our attention on 
the first circumstance, the concept of force appears as 
the original notion, the concept of work as the derived 
one. If we investigate the influence of the second fact 
first, the concept of work is the original notion. In 
the transference of the ideas reached in the observation 
of the motion of descent to more complicated relations, 
force is recognised as dependent on the distance be- 
tween the bodies that is, as a function of the distance, 
f{f). The work done through the element of distance dr 
is then/(r) dr. By the second method of investiga- 
tion work is also obtained as a function of the distance, 
F {/) ; but in this case we know force only in the form 
d. F {f)ldr that is to say, as the limiting value of the 
ratio : (increment of work)/(increment of distance.) 

The prefer- Galileo cultivated by preference the first of these 

cnccs of the 

different in- two methods. Newton likewise preferred it. Huygens 
pursued the second method, without at all restricting 
himself to it. Descartes elaborated Galileo's ideas after 
a fashion of his own. But his performances are in- 
significant compared with those of Newton and Huy- 
gens, and their influence was soon totally effaced. After 
Huygens and Newton, the mingling of the two spheres 
of thought, the independence and equivalence of which 
are not always noticed, led to various blunders and 
confusions, especially in the dispute between the Car- 
tesians and LeibniJfeians, already referred to, concern- 
ing the measure 01 force. In recent times, however, in- 
quirers turn by preference now to the one and now to 



THE PRINCIPLES OF D YXAMICS. 251 

the other. Thus the Galileo- Newtonian ideas are culti- 
vated with preference by the school of Poinsot, the 
Galileo-Huygenian by the school of Poncelet. 

4. Newton operates almost exclusively with the no- The impor- 

' r tance and 

tions of force, mass, and momentum. His sense of the history of 

the New- 
Value of the concept of mass places him above his prede- toman con- 

... z 1-1 cept * 

cessors and contemporaries. It did not occur to Galileo mass. 

that mass and weight were different things. Huygens, 
too, in all his considerations, puts weights for masses ; 
as for example in his investigations concerning the 
centre of oscillation. Even in the treatise De Percus- 
sione (On Impact), Huygens always says "corpus ma- 
jus," the larger body, and "corpus minus," the smaller 
body, when he means the larger or the smaller mass. 
Physicists were not led to form the concept mass till 
they made the discovery that the same body can by the 
action of gravity receive different accelerations. The 
first occasion of this discovery was the pendulum-ob- 
servations of Richer (1671-1673), from which Huy- 
gens at once drew the proper inferences, and the 
second was the extension of the dynamical laws to the 
heavenly bodies. The importance of the first point may 
be inferred from the fact that Newton, to prove the pro- 
portionality of mass and weight on the same spot of the 
earth, personally instituted accurate observations on 
pendulums of different materials (Principia. Lib. II, 
Sect. VI, De Motu et Resistentia Corporum Funependu- 
loruiti). In the case of John Bernoulli, also, the first 
distinction between mass and weight (in the Meditatio 
de Xatura Centri Oscillationis. Opera Omnia, Lausanne 
and Geneva, Vol. II, p. 168) was made on the ground 
of the fact that the same body can receive different 
gravitational accelerations. Newton, accordingly, dis- 
poses of all dynamical questions involving the relations 



252 THE SCIENCE OF MECHANICS. 

of several bodies to each other, by the help of the ideas 
of force, mass, and momentum. 
Themeth- 5. Huygens pursued a different method for the so- 

ods of Huy- . . . . , , _, , , ' .... 

gens. iution 01 these problems. Galileo had previously dis- 

covered that a body rises by virtue of the velocity ac- 
quired in its descent to exactly the same height as that 
from which it fell. Huygens, generalising the principle 
(in his Horologium Oscillatoriuni) to the effect that the 
centre of gravity of any system of bodies will rise by 
virtue of the velocities acquired in its descent to ex- 
actly the same height as that from which it fell, reached 
the principle of the equivalence of work and vis viva. 
The names of the formulae which he obtained, were, 
of course, not supplied until long afterwards. 

The Huygenian principle of work was received by 
the contemporary world with almost universal distrust. 
People contented themselves with making use of its 
brilliant consequences. It was always their endeavor 
to replace its deductions by others. Even after John 
and Daniel Bernoulli had extended the principle, it 
was its fruitfulness rather than its evidency that was 
valued. 
Themeth- We observe, that the Galileo-Newtonian principles 
ton and were, on account of their greater simplicity and ap- 
compared. parently greater evidency, invariably preferred to the 
Galileo-Huygenian. The employment of the latter is 
exacted only by necessity in cases in which the em- 
ployment of the former, owing to the laborious atten- 
tion to details demanded, is impossible ; as in the case 
of John and Daniel Bernoulli's investigations of the 
motion of fluids. 

If we look at the matter closely, however, the same 
simplicity and evidency will be found to belong to the 
Huygenian principles as to the Newtonian proposi- 



THE PRINCIPLES OF DYNAMICS. 253 

tions. That the velocity of a body is determined by 
the time of descent or determined by the distance of 
descent, are assumptions equally natural and equally 
simple. The form of the law must in both cases be 
supplied by experience. As a starting-point, therefore, 
pt = mv and ps = mr 2 /2 are equally well fitted. 

6. When we pass to the investigation of the motion The neces- 

* . sity and 

of several bodies, we are again compelled, in both cases, umversai- 

' r . ity of the 

to take a second step of an equal degree of certainty, two meth- 

ods. 

The Newtonian idea of mass is justified by the fact, 
that, if relinquished, all rules of action for events would 
have an end ; that we should forthwith have to expect 
contradictions of our commonest and crudest experi- 
ences ; and that the physiognomy of our mechanical 
environment would become unintelligible. The same 
thing must be said of the Huygenian principle of work. 
If we surrender the theorem 2ps = 2mv 2 /2, heavy 
bodies will, by virtue of their own weights, be able to 
ascend higher ; all known rules of mechanical occur- 
rences will have an end. The instinctive factors which 
entered alike into the discovery of the one view and of 
the other have been already discussed. 

The two spheres of ideas could, of course, have The points 

1 5 1/11. T> ' COntaCt 

grown up much more independently of each other. But of the two 

r r J methods. 

in view of the fact that the two were constantly in con- 
tact, it is no wonder that they have become partially 
merged in each other, and that the Huygenian appears 
the less complete. Newton is all-sufficient with his 
forces, masses, and momenta. Huygens would like- 
wise suffice with work, mass, and vis viva. But since 
he did not in his time completely possess the idea of 
mass, that idea had in subsequent applications to be 
borrowed from the other sphere. Yet this also could 
have been avoided. If with Newton the mass-ratio of 



254 THE SCIENCE OF MECHANICS. 

two bodies can be defined as the inverse ratio of the 
velocities generated by the same force, with Huygens 
it would be logically and consistently definable as the 
inverse ratio of the squares of the velocities generated 
by the same work. 
The respec- The two spheres of ideas consider the mutual de- 

tive merits 

of each. pendence on each other of entirely different factors of 
the same phenomenon. The Newtonian view is in so 
far more complete as it gives us information regarding 
the motion of each mass. But to do this it is obliged 
to descend greatly into details. The Huygenian view 
furnishes a rule for the whole system. It is only a con- 
venience, but it is then a mighty convenience, when 
the relative velocities of the masses are previously and 
independently known. 

The gen- 7. Thus we are led to see, that in the develop- 

eral devel- . r 

opment of ment of dynamics, just as in the development of statics, 

dynamics ... 

in the light the connection of widely different features of mechanical 

of the pre- 
ceding re- phenomena engrossed at different times the attention 

marks. ... 

of inquirers. We may regard the momentum of a sys- 
tem as determined by the forces ; or, on the other 
hand, we may regard its vis viva as determined by the 
work. In the selection of the criteria in question the 
individuality of the inquirers has great scope. It will 
be conceived possible, from the arguments above pre- 
sented, that our system of mechanical ideas might, 
perhaps, have been different, had Kepler instituted 
the first investigations concerning the motions of fall- 
ing bodies, or had Galileo not committed an error in 
his first speculations. We shall recognise also that not 
only a knowledge of the ideas that have been accepted 
and cultivated by subsequent teachers is necessary for 
the historical understanding of a science, but also that 
the rejected and transient thoughts of the inquirers, 



THE PRISCIPLES OF DYNAMICS. 255 

nay even apparently erroneous notions, may be very 
important and very instructive. The historical investi- 
gation of the development of a science is most needful, 
lest the principles treasured up in it become a system 
of half-understood prescripts, or worse, a system of 
prejudices. Historical investigation not only promotes 
the understanding of that which now is, but also brings 
;new possibilities before us, by showing that which ex- 
ists to be in great measure conventional and accidental. 
From the higher point of view at which different paths 
! of thought converge we may look about us with freer ^"jf?' f 

ipowers of vision and discover routes before unknown. 

In all the dvnamical propositions that we have dis- The substi- 

1 a 1 <-r>i tution of 

cussed, velocity plays a prominent role. The reason integral 

,...". ., , . , , for "differ- 

01 this, in our view, is, that, accurately considered, entiai" 

1 t 1 r ' aws ma y 

every single body of the universe stands in some den- some day 

IlKikc ttl6 

lite relation with every other body in the universe ; concept of 

1 111 11 force super 

that any one body, and consequently also any several fluous. 
bodies, cannot be regarded as wholly isolated. Our 
inability to take in all things at a glance alone compels 
;us to consider a few bodies and for the time being to 
/ in certain aspects the others ; a step accom- 
plished by the introduction of velocity, and therefore 
of time. We cannot regard it as impossible that inte- 
gral laws, to use an expression of C. Neumann, will 
some day take the place of the laws of mathematical 
elements, or differential laws, that now make up the 
science of mechanics, and that we shall have direct 
knowledge of the dependence on one another of the 
Positions of bodies. In such an event, the concept of 
force will have become superfluous. (See Appendix, 
XXI., p. 548, on Hertz's Mechanics ; also Appendix 
XXII., p. 555, in answer to criticisms of the views ex- 
pressed by the author in Chapters I. and II.) 



CHAPTER III. 



Newton's 
principles 
are uni- 
versal in 
scope and 
power. 



THE EXTENDED APPLICATION OF THE PRINCIPLES 
OF MECHANICS AND THE DEDUCTIVE DE- 
VELOPMENT OF THE SCIENCE. 



SCOPE OF THE NEWTONIAN PRINCIPLES. 






i. The principles of Newton suffice by themselves, 
without the introduction of any new laws, to explore 
thoroughly every mechanical phenomenon practically 
occurring, whether it belongs to statics or to dynamics. 
If difficulties arise in any such consideration, they are 
invariably of a mathematical, or 
formal, character, and in no re- 
spect concerned with questions 
of principle. We have given, 
let us suppose, a number of mas- 
ses m y , m 2 , m % . . . .in space, with 
definite initial velocities v lf v 2 , 
7' 3 . . . . We imagine, further, lin 
of junction drawn between eve 
fk- 144- two masses. In the directions 

these lines of junction are set up the accelerations an 
counter-accelerations, the dependence of which on th 
distance it is the business of physics to determine. I 
a small element of time r the mass m s , for example, 
will traverse in the direction of its initial velocity the 
distance v.r, and in the directions of the lines joining 




THE EXTENSION OF THE PRIXCIPLES. 257 

it with the masses m., m , m . . . ., being affected in Schematic 

... . -ii " i - c - illustration 

such directions with the accelerations cp, cp%, cp. . . ., of the pre- 

ceding 
the distances ((p\/2)T 2 , (q>\/2)T 2 , (<p|/2)r 2 . . . . If statement. 

we imagine all these motions to be performed indepen- 
dently of each other, we shall obtain the new position 
of the mass m. after lapse of time r. The composition 
of the velocities r 5 and (p\T, <p\t, <p\* gives the 
new initial velocity at the end of time r. We then 
allow a second small interval of time r to elapse, and, 
making allowance for the new spatial relations of the 
masses, continue in the same way the investigation of 
the motion. In like manner we may proceed with 
every other mass. It will be seen, therefore, that, in 
point of principle, no embarrassment can arise ; the 
difficulties which occur are solely of a mathematical 
character, where an exact solution in concise symbols, 
and not a clear insight into the momentary workings 
of the phenomenon, is demanded. If the accelerations 
of the mass ///., or of several masses, collectively neu- 
tralise each other, the mass m. or the other masses 
mentioned are in equilibrium and will move uniformly 
onwards with their initial velocities. If, in addition, 
: the initial velocities in question are = 0, both equilib- 
rium and rest subsist for these masses. 

Nor, where a number of the masses m , ///., .... The same 

1 . . idea ap- 

have considerable extension, so that it is impossible to pHed to ag- 

,,.,..... . , , . , Bregates of 

speak ot a single line joining every two masses, is the dif- material 
nculty, in point of principle, any greater. We divide 
the masses into portions sufficiently small for our pur- 
pose, and draw the lines of junction mentioned between 
every two such portions. We, furthermore, take into 
account the reciprocal relation of the parts of the 
same large mass ; which relation, in the case of rigid 
masses for instance, consists in the parts resisting 



258 THE SCIENCE OF MECHANICS. 






every alteration of their distances from one another 
On the alteration of the distance between any two parts 
of such a mass an acceleration is observed proportional 
to that alteration. Increased distances diminish, and 
diminished distances increase in consequence of this 
acceleration. By the displacement of the parts with 
respect to one another, the familiar forces of elasticity 
are aroused. When masses meet in impact, their 
forces of elasticity do not come into play until contact 
and an incipient alteration of form take place. 
a practical 2. If we imagine a heavy perpendicular column 
of thescope resting on the earth, any particle m in the interior of 
princ? P iTs. s the column which we may choose to isolate in thought, 
is in equilibrium and at rest. A vertical downward ac- 
celeration g is produced by the earth in the particle, 
which acceleration the particle obeys. But in so doing 
it approaches nearer to the particles lying beneath it, 
and the elastic forces thus awakened generate in ;// a 
vertical acceleration upwards, which ultimately, when 
the particle has approached near enough, becomes 
equal to g. The particles lying above m likewise 
approach m with the acceleration g. Here, again, 
acceleration and counter-acceleration are produced, 
whereby the particles situated above are brought to j 
rest, but whereby m continues to be forced nearer and 
nearer to the particles beneath it until the acceleration | 
downwards, which it receives from the particles above ; 
it, increased by g, is equal to the acceleration .it re- 
ceives in the upward direction from the particles be- j 
neath it. We may apply the same reasoning to every 
portion of the column and the e'arth beneath it, readily, 
perceiving that the lower portions lie nearer each other: 
and are more violently pressed together than the parts 
above. Every portion lies between a less closely pressed 






THE EXTENSION OF THE PRIXCIPLES. 259 

upper portion and a more closely pressed lower por- Rest in the 
tion ; its downward acceleration g is neutralised by a these prin- 
surplus of acceleration upwards, which it experiences pears as a 
from the parts beneath. We comprehend the equilib- of motion, 
rium and rest of the parts of the column by imagining 
all the accelerated motions which the reciprocal rela- 
tion of the earth and the parts of the column determine, 
as in fact simultaneously performed. The apparent 
mathematical sterility of this conception vanishes, and 
it assumes at once an animate form, when we reflect 
that in reality no body is completely at rest, but that 
in all, slight tremors and disturbances are constantly 
taking place which now give to the accelerations of de- 
scent and now to the accelerations of elasticity a slight 
preponderance. Rest, therefore, is a case of motion, 
very infrequent, and, indeed, never completely realised. 
The tremors mentioned are by no means an unfamiliar 
phenomenon. When, however, we occupy ourselves 
with cases of equilibrium, we are concerned simply with 
a schematic reproduction in thought of the mechanical 
facts. We then purposely neglect these disturbances, 
displacements, bendings, and tremors, as here they 
have no interest for us. All cases of this class, which 
have a scientific or practical importance, fall within the 
province of the so-called theory of elasticity. The whole The unity 

. . and homo- 

Outcome of Newton's achievements is that we every- gemHty 

where reach our goal with one and the same idea, and principles 

, r . , t 5 introduce 

oy means 01 it are able to reproduce and construct be- into the 

7 , .. . ..... , science. 

iorenand all cases of equilibrium and motion. All 
phenomena of a mechanical kind now appear to us 
as uniform throughout and as made up of the same 
elements. 

3. Let us consider another example. Two mas- 
ses m, m are situated at a distance a from each 



2 6o THE SCIENCE OE MECHANICS. 

a general other. (Fig. 1 45.) When displaced with respect to 

ixemplifi- , , . . . , . . 

cation of each other, elastic forces proportional to the change 
oftheprin- x 2 of distance are supposed to be 

E3 g V awakened. Let the masses be 

movable in the X-direction par- 
allel to a, and their coordinates 
be x x , x 2 . If a force /is applied at the point x 2 , the 
following equations obtain : 

d- x 
'" lt^ = ^^ x 2 x i) \ C 1 ) 



d 2 x 



in 



._/[(*,_ x x ) - a] +/ (2) 

where p stands for the force that one mass exerts on 
the other when their mutual distance is altered by the 
value 1. All the quantitative properties of the me- 
chanical process are determined by these equations. 
But we obtain these properties in a more comprehensi- 
ble form by the integration of the equations. The ordi- 
nary procedure is, to find by the repeated differentia- 
tion of the equations before us new equations in suffi- 
cient number to obtain by elimination equations in x x 
alone or r 2 alone, which are afterwards integrated. We 
shall here pursue a different method. By subtracting 
the first equation from the second, we get 

The dervel- m ^ ? ~ - 1 - = 2/[(.r x ) a] -j-/, Or 

opment of at - 



putting x % X x = 1/, 



the equa- 
tions ob- 
tained in 
this exam- , 

p,e - ^==-2/J*-]+/ (3) 

and by the addition of the first and the second equa- 
tions 

d 2 (x 2 -4- x-l) 



_-/ or, putting .r 2 -f- x x = 






rilE EXTENSION OF THE PRINCIPLES. 



261 



m jr*=f 



(4) 



The integrals of (3) and (4) are respectively 

// = A sin x i"^ . t + B cos x p . / -f a -f- L. and 
> m \ /// 2/ 



The integ- 
rals of these 
develop- 
ments. 



- 4- Ct 4- D; whence 
/// 2 



A . \l'p B - f f* 

'1 = ., sin \ ' T cos \h ' -r :, ., 

2 \ ill 2 \ ;// 1m 2 



+-i-i+T 



= = sinJ J ./+ ., cos J -A /-f .; 



2m 2 



a f D 

+ c ' + 2+4+ r 

To take a particular case, we will assume that the a partica- 

lar case of 

action of the force/ begins at /= 0, and that at this the \am- 

ple. 

time 



*, 



</.V, 



that is, the initial positions are given and the initial 
velocities are = 0. The constants A, B, C, D being 
eliminated by these conditions, we get 



r ^4/ C< 



I 



^ tm ' 2 4/ 



(6) x, 

(7) x % 



{- cos J "^ . / -f / / r + a + /, an 
\p \ in ' 2/// 2 ' 7 4/ 



5 \li 

/ |2> / 

2/ \ m T 2/ 



262 THE SCIENCE OF MECHANICS. 

Theinform- We see from (5) and (6) that the two masses, in addi- 

ation which . . J y 

the result- tion to a uniformly accelerated motion with half the 

ant equa- 
tions give acceleration that the force/ would impart to one of 

concerning 

this exam- these masses alone, execute an oscillatory motion sym- 
metrical with respect to their centre of gravity. The 
duration of this oscillatory motion, T= 2 nVm/ip, is 
smaller in proportion as the force that is awakened in 
the same mass-displacement is greater (if our attention 
is directed to two particles of the same body, in pro- 
portion as the body is harder). The amplitude of os- 
cillation of the oscillatory motion f/^p likewise de- 
creases with the magnitude/ of the force of displace- 
ment generated. Equation (7) exhibits the periodic 
change of distance of the two masses during their pro- 
gressive motion. The motion of an elastic body might 
in such case be characterised as vermicular. With hard 
bodies, however, the number of the oscillations is so 
great and their excursions so small that they remain 
unnoticed, and may be left out of account. The oscil- 
latory motion, furthermore, vanishes, either gradually 
through the effect of some resistance, or when the two 
masses, at the moment the force /begins to act, are a 
distance a -\-fJ7.p apart and have equal initial veloci- 
ties. The distance a -f- f/zp that the masses are apart 
after the vanishing of their vibratory motion, is//2/ 
greater than the distance of equilibrium a. A tension 
j, namely, is set up by the action of/, by which the 
acceleration of the foremost mass is reduced to one- 
half whilst that of the mass following is increased by 
the same amount. In this, then, agreeably to our as- 

This in- sumption, pyltn =//% m or y =flip. As we see, it is 

formation . . . , ., 

isexhaus- in our power to determine the minutest details of a 
phenomenon of this character by the Newtonian prin- 
ciples. The investigation becomes (mathematically, 



THE EXTEXSIOX OF THE PRIXCIPLES. 263 

yet not in point of principle) more complicated when 
we conceive a body divided up into a great number of 
small parts that cohere by elasticity. Here also in the 
case of sufficient hardness the vibrations may be neg- 
lected. Bodies in which we purposely regard the mu- 
tual displacement of the parts as evanescent, are called 
bodies. 

4. We will now consider a case that exhibits the The dednc- 
schema of a lever. We imagine the masses M y m lt m 2 laws of the 
arranged in a triangle and joined by elastic connec- Newton's 
tions. Even* alteration of the sides, and consequently 
also every alteration of the angles, gives rise to accel- 
erations, as the result of which the triangle endeavors to 
assume its previous form and size. By the aid of the 
Newtonian principles we can deduce from such a 
schema the laws of the lever, and at the same time feel 
that the form of the deduction, although it may be 
more complicated, still 
remains admissible when 
we pass from a schematic 
lever composed of three 
masses to the case of a 
real lever. The mass J/ 

we assume either to be in itself very large or conceive 
it joined by powerful elastic forces to other very large 
masses (the earth for instance). M then- represents 
an immovable fulcrum. 

Let m x , now. receive from the action of some ex- The 
ternal force an acceleration f perpendicular to the line deduction 
of junction Mm 2 = c -f- d. Immediately a stretching 
of the lines m v m 2 =6 and m x M = a is produced, and 
in the directions in question there are respectively set 
up the accelerations, as yet undetermined, s and G y of 
which the components s(e/fi) and <r(e/a) are directed 




Fig. 146. 



264 THE SCIENCE OF MECHANICS. 

oppositely to the acceleration/. Here e is the altitude 
of the triangle vi x tn 2 M. The mass m 2 receives the 
acceleration s', which resolves itself into the two com- 
ponents s'(d//>) in the direction of M and s'(e/b) par- 
allel to f. The former of these determines a slight ap- 
proach of m 2 to M. The accelerations produced in M 
by the reactions of m. and m 2 , owing to its great mass, 
are imperceptible. We purposely neglect, therefore, 
the motion of M. 
The deduc- The mass m., accordingly, receives the accelera- 
tainedbv tion / s(e/b) G{ejd), whilst the mass m 2 suffers 
tration of the parallel acceleration s\e/b). Between s and G a 
dons. simple relation obtains. Ifj by supposition, we have a 

very rigid connection, the triangle is only impercept- 
ibly distorted. The components of s and G perpendicular 
to / destroy each other. For if this were at any one 
moment not the case, the greater component would 
produce a further distortion, which would immediately 
counteract its excess. The resultant of s and G is 
therefore directly contrary to/, and consequently, as is 
readily obvious, G (c/a) = s (d/b). Between s and s', 
further, subsists the familiar relation m 1 s = m 2 s' or 
s = s'(m 2 /m 1 ). Altogether m 2 and /// 1 receive re- 
spectively the accelerations s'{e //>) and / s'{e/b) 
(jn 2 / m i)(. c ~^~ d/t)> or, introducing in the place of the 
variable value s'iejb') the designation cp, the accelera- 
tions <p and/ cp(m 2 /m 1 } (V -f- d/e). 
On the pre- At the commencement of the distortion, the accel- 

ceding sup- . r . . . e , , 

positions eration of m lt owing to the increase of <p, diminishes, 
the rotation whilst that of ? 2 increases. If we make the altitude e 

of the lever . , . , ,. .,, 

are easily of the triangle very small, our reasoning still remains 
applicable. In this case, however, a becomes = c == r v 
and a -f- b = c-\-d=r 2 . We see, moreover, that the 
distortion must continue, cp increase, and the accelera- 



THE EXTEXSIOX Of THE PRINCIPLES. 265 

tion of m Y diminish until the stage is reached at which 
the accelerations of m x and m t bear to each other the 
proportion of r, to r 2 . This is equivalent to a rotation 
of the whole triangle (without further distortion) about 
M, which mass by reason of the vanishing accelera- 
tions is at rest. As soon as rotation sets in, the rea- 
son for further alterations of <p ceases. In such a case, 
consequently, 

M f m 2 r i\ r \ m \f 

<P= - ! / <p -3 '--3 } or q> = r, -* -= -. 

r t ( m l r l ] - m % r % * + / 3 r 2 * 

For the angular acceleration f of the lever we get 
<p _ r x mj 






! 



m x r x * + m 2 



Nothing prevents us from entering still more into Discussion 

of the char- 

the details of this case and determining the distortions acter of the 
and vibrations of the parts with respect to 'each other, result. 
With sufficiently rigid connections, however, these de- 
tails may be neglected. It will be perceived that we 
have arrived, by the employment of the Newtonian prin- 
ciples, at the same result to which the Huygenian view 
also would have led us. This will not appear strange to 
us if we bear in mind that the two views are in every re- 
spect equivalent, and merely start from different aspects 
of the same subject-matter. If we had pursued the 
Huygenian method, we should have arrived more 
speedily at our goal but with less insight into the de- 
tails of the phenomenon. We should have employed 
the work done in some displacement of m J to deter- 
mine the vires vizue of m y and w,, wherein we should 
have assumed that the velocities in question v x , v., 
maintained the ratio v 1 /v a =r t /r a . The example 
here treated is very well adapted to illustrate what 
such an equation of condition means. The equation 



266 



THE SCIENCE OF MECHANICS. 



simply asserts, that on the slightest deviations of v 1 /v 2 
from r 1 /r 2 powerful forces are set in action which in 
point of fact prevent all further deviation. The bodies 
obey of course, not the equations, but the forces. 
a simple 5. We obtain a very obvious case if we put in the 

case of the . . . /T -,. 

same exam- example just treated m l =m 2 =m and a = o(big. 
147). The dynamical state of the system ceases to 
change when <p = 2 (/ 2 cp), that is, when the accel- 
erations of the masses 
at the base and the ver- 
tex are given by 2//5 
and f/$. At the com- 
mencement of the dis- 
tortion cp increases, and simultaneously the accelera- 
tion of the mass at the vertex is decreased by double 
that amount, until the proportion subsists between the 
two of 2 : 1. 
Theequi- We have yet to consider the case of equilibrium of 

librium of , . , . . ._. ON , . 

the lever a schematic lever, consisting (Fig. 40) of three masses 
from the m , ;//, and M, of which the last is again supposed 

same con- 
siderations. 




Fig- M7- 




Fig. 148. 

to be very large or to be elastically connected with 
very large masses. We imagine two equal and oppo- 
site forces s, s applied to m t and m 2 in the direction 
m 1 m 2 , or, what is the same thing, accelerations im- 
pressed inversely proportional to the masses m r , m 2 . 
The stretching of the connection m x m % also generates 



THE EXTEXSIOX OF THE PRIXCIPLES. 267 

accelerations inversely proportional to the masses m, s 
?n 2 , which neutralise the first ones and produce equi- 
librium. Similarly, along m x M imagine the equal and 
contrary forces /, / operative ; and along m 2 M the 
forces //, //. In this case also equilibrium obtains. 
If M be elastically connected with masses sufficiently 
large, u and / need not be applied, inasmuch 
as the last-named forces are spontaneously evoked the 
moment the distortion begins, and always balance the 
forces opposed to them. Equilibrium subsists, accord- 
ingly, for the two equal and opposite forces s, s as 
well as for the wholly arbitrary forces /, u. As a matter 
of fact s, s destroy each other and /, u pass through 
the fixed mass M, that is, are destroyed on distortion 
setting in. 

The condition of equilibrium readily reduces itself The rednc- 
to the common form when we reflect that the mo- preceding 
ments of / and u, forces passing through M, are with common 
respect to M zero, while the moments of s and s are 
equal and opposite. If we compound / and s to p, and 
u and s\o q, then, by Varignon's jw^/r/V<z/ principle 
of the parallelogram, the moment of / is equal to the 
sum of the moments of s and /, and the moment of q 
is equal to the sum of the moments of u and s. The 
moments of / and q are therefore equal and opposite. 
Consequently, any two forces / and q will be in equi- 
librium if they produce in the direction m x m 2 equal 
and opposite components, by which condition the equal- 
ity of the moments with respect to M is posited. That 
then the resultant of / and q also passes through M, is 
likewise obvious, for s and * destroy each other and 
/ and u pass through M. 

6. The Newtonian point of view, as the example 
just developed shows us, includes that of Varignon. 



268 THE SCIEXCE OF MECHANICS. 

Newton's We were right, therefore, when we characterised the 

point of . . 

view in- statics of Vangnon as a dynamical statics, which, start- 

cludes . " _ 

Varignon't. ing from the fundamental ideas of modern dynamics, 
voluntarily restricts itself to the investigation of cases 
of equilibrium. Only in the statics of Varignon, owing 
to its abstract form, the significance of many opera- 
tions, as for example that of the translation of the 
forces in their own directions, is not so distinctly ex- 
hibited as in the instance just treated. 

Theecon- The considerations here developed will convince 

omy and . ... 

wealth of us that we can dispose by the Newtonian principles 
ian ideas, of every phenomenon of a mechanical kind which may 
arise, provided we only take the pains to enter far 
enough into details. We literally see through the cases 
of equilibrium and motion which here occur, and be- 
hold the masses actually impressed with the accelera- 
tions they determine in one another. It is the same 
grand fact, which we recognise in the most various 
phenomena, or at least can recognise there if we make 
a point of so doing. Thus a unity, homogeneity, and 
economy of thought were produced, and a new and 
wide domain of physical conception opened which 
before Newton's time was unattainable. 
The New- Mechanics, however, is not altogether an end in it- 

themodern, self ; it has also problems to solve that touch the needs 
methods, of practical life and affect the furtherance of other sci- 
ences. Those problems are now for the most part ad- 
vantageously solved by other methods than the New- 
tonian, methods whose equivalence to that has already 
been demonstrated. It would, therefore, be mere im- 
practical pedantry to contemn all other advantages and 
insist upon always going back to the elementary New- 
tonian ideas. It is sufficient to have once convinced 
ourselves that this is always possible. Yet the New- 



THE EXTENSION OF THE PRIXCIPLES. 269 

tonian conceptions are certainly the most satisfactory 
and the most lucid ; and Poinsot shows a noble sense 
of scientific clearness and simplicity in making these 
conceptions the sole foundation of the science. 



THE FORMULAE AND UNITS OF MECHANICS. 

i. All the important formulae of modern mechanics History of 

._ the fortnu- 

were discovered and employed in the period of Galileo i and 

. . . . units of 

and Newton. The particular designations, which, mechanics. 
owing to the frequency of their use, it was found con- 
venient to give them, were for the most part not fixed 
upon until long afterwards. The systematical mechan- 
ical units were not introduced until later still. Indeed, 
the last named improvement, cannot be regarded as 
having yet reached its completion. 

2. Let s denote the distance, /the time, v the in-Theorig- 
stantaneous velocity, and q> the acceleration of a uni- tlons*??*" 
formly accelerated motion. From the researches of Huygens" 1 ' 
Galileo and Huygens, we derive the following equa- 
tions : 

v = tpt 



9 s = 9 



(1) 



Multiplying throughout by the mass m, these equa- The intro- 
tions give the following : or- mas 

and " mov- 
/// : /// cpt in 8 force." 



mv 



m <ps=^ 



270 THE SCIENCE OF MECHANICS 

Final form and, denoting the moving force m at by the letter />, we 

of the fun- . ' J J ' 

damental obtain 

equations. 



mv =pt 

pt 2 



ps 



;2) 



Equations (i) all contain the quantity cp ; and each 
contains in addition two of the quantities s, t, v, as 
exhibited in the following table : 

(v,t 
<p \ s, t 
{ s, V 
Equations (2) contain the quantities m, p, s, t, v ; 
each containing ;//, p and in addition to m, p two of the 
three quantities s, t, v, according to the following table : 

r v, t 

m,p j s, t 
The scope Questions concerning motions due to constant forces 

and appli- . , . , . 

cation of are answered by equations (2) in great variety. If, for 

these equa- . . , , . , 

tions. example, we want to know the velocity v that a mass 
m acquires in the time / through the action of a force 
/, the first equation gives v ^pf/m. If, on the other 
hand, the time be sought during which a mass ;;/ with 
the velocity v can move in opposition to a force/, the 
same equation gives us t = m v/p. Again, if we in- 
quire after the distance through which m will move with 
velocity v in opposition to the force/, the third equa- 
tion gives s = mv 2 /2p. The two last questions illus- 
trate, also, the futility of the Descartes-Leibnitzian dis- 
pute concerning the measure of force of a body in mo- 
tion. The use of these equations greatly contributes 



THE EXTENSION OF THE PRIXCIPLES. 271 

to confidence in dealing with mechanical ideas. Sup- 
pose, for instance, we put to ourselves, the question, 
what force p will impart to a given mass m the velocity 
v ; we readily see that between m, p, and v alone, no 
equation exists, so that either s or / must be supplied, 
and consequently the question is an indeterminate one. 
We soon learn to recognise and avoid indeterminate 
cases of this kind. The distance that a mass /// acted 
on by the force / describes in the time /, if moving 
with the initial velocity 0, is found by the second equa- 
tion s =pt 2 /2m. 

3. Several of the formulae in the above-discussed The names 

which the 

equations have received particular names. The force formulas of 
of a moving body was spoken of by Galileo, who al- tionshave 
ternately calls it "momentum," "impulse," and "en- 
ergy." He regards this momentum as proportional to 
the product of the mass (or rather the weight, for Gali- 
leo had no clear idea of mass, and for that matter no 
more had Descartes, nor even Leibnitz) into the velo- 
city of the body. Descartes accepted this view. He put 
the force of a moving body mv, called it quantity of 
motion, and maintained that the sum-total of the quan- 
tity of motion in the universe remained constant, so that 
when one body lost momentum the loss was compen- 
sated for by an increase of momentum in other bodies. 
Newton also employed the designation " quantity of 
motion " for /// v, and this name has been retained to the Momen- 

. , __-, , , _ turn and 

present day. [But momentum is the more usual term. J impulse. 
For the second member of the first equation, viz. //, 
Belanger, proposed, as late as 1847, the name impulse* 
The expressions of the second equation have received 

* See, also, Maxwell, Matter and Motion, American edition, page 72. But 
this word is commonly used in a different sense, namely, as "the limit of a 
force which is infinitely great but acts only during an infinitely short time." 
See Routh, Rigid Dynamics, Part I. pages 65-66. Tram. 



272 . THE SCIENCE OF MECHANICS. 

Sis viva no particular designations. Leibnitz (1605) called the 

and work. . # 

expression mv- of the third equation vis viva or firing 
force, and he regarded it, in opposition to Descartes, 
as the true measure of the force of a body in motion, 
calling the pressure of a body at rest vis morlua, or 
dead force. Coriolis found it more appropriate to give 
the term %mv 2 the name vis viva. To avoid confusion, 
Belanger proposed to call;//?' 2 living force andiwr 2 
living power [now commonly called in English kinetic 
energy\. For ps Coriolis employed the name work. 
Poncelet confirmed this usage, and adopted the kilo- 
gramme-metre (that is, a force equal to the weight of a 
kilogramme acting through the distance of a metre) as 
the unit of work. 
The history a Concerning the historical details of the origin of 

of the ideas ^ . , . . 

quantity of these notions "quantity of motion" and "vis viva," 

motion and . . . 

vis viva, a glance may now be cast at the ideas which led Des- 
cartes and Leibnitz to their opinions. In his Principia 
Philosophic, published in 1644, II, 36, Descartes ex'- 
pressed himself as follows : 

" Now that the nature of motion has been examined, 
"we must consider its cause, which may be conceived 
"in two senses : first, as a universal, original cause 
"the general cause of all the motion in the world ; and 
"second, as a special cause, from which the individual- 
" parts of matter receive motion which before they did 
"not have. As to the universal cause, it can mani- 
festly be none other than God, who in the beginning 
" created matter with its motion and rest, and who now 
" preserves, by his simple ordinary concurrence, on the 
"whole, the same amount of motion and rest as he 
"originally created. For though motion is only a con- 
dition of moving matter, there yet exists in matter 
" a definite quantity of it, which in the world at large 



THE EXTEXSIOX OF THE PRIXCIPLES. 273 

' never increases or diminishes, although in single por- Passage 
' tions it changes ; namely, in this way, that we must cartefi 
1 assume, in the case of the motion of a piece of matter 
' which is moving twice as fast as another piece, but in 
'quantity is only one half of it, that there is the same 
' amount of motion in both, and that in the proportion 
' as the motion of one part grows less, in the same pro- 
' portion must the motion of another, equally large 
'part grow greater. We recognise it, moreover, as 
'a perfection of God, that He is not only in Himself 
' unchangeable, but that also his modes of operation 
'are most rigorous and constant ; so that, with the ex- 
ception of the changes which indubitable experience 
'or divine revelation offer, and which happen, as our 
'faith or judgment show, without any change in the 
' Creator, we are not permitted to assume any others 
' in his works lest inconstancy be in any way pre- 
dicated of Him. Therefore, it is wholly rational to 
'assume that God, since in the creation of matter he 
' imparted different motions to its parts, and preserves 
'all matter in the same way and conditions in which 
'he created it, so he similarly preserves in it the same 
1 quantity of motion" (See Appendix, XXIII., p. 574.) 

The merit of having first sought after a more uni- The menu 

.... and defects 

versal and more fruitful point of view in mechanics, of Descar- 

I, ,ess pfcys- 

cannot be denied Descartes. This is the peculiar task icaiinquir- 

of the philosopher, and it is an activity which con- 
stantly exerts a fruitful and stimulating influence on 
physical science. 

Descartes, however, was infected with all the usual 
errors of the philosopher. He places absolute confi- 
dence in his own ideas. He never troubles himself to 
put them to experiential test. On the contrary, a min- 
imum of experience always suffices him for a maximum 



274 THE SCIENCE OF MECHANICS. 

of inference. Added to this, is the indistinctness of 
his conceptions. Descartes did not possess a clear 
idea of mass. It is hardly allowable to say that Des 
cartes defined mv as momentum, although Descartes's 
scientific successors, feeling the need of more definite 
notions, adopted this conception. Descartes's greatest 
error, however,: and the one that vitiates all his phys- 
ical inquiries, is this, that many propositions appear 
to him self-evident a priori concerning the truth of 
which experience alone can decide. Thus, in the two 
paragraphs following that cited above (37-39) it is 
asserted as a self-evident proposition that a body pre- 
serves unchanged its velocity and direction. The ex- 
periences cited in 38 should have been employed, not 
as a confirmation of an a priori law of inertia, but as a 
foundation on which this law in an empirical sense 
should be based. 
Leibnitz Descartes's view was attacked by Leibnitz (1686) 

on quantity p . . * 

of motion, in the Acta Eruditorum, in a little treatise bearing the 
title : " A short Demonstration of a Remarkable Error 
of Descartes and Others, Concerning the Natural Law 
by which they think that the Creator always preserves 
the same Quantity of Motion ; by which, however, the 
Science of Mechanics is totally perverted." 

In machines in equilibrium, Leibnitz remarks, the 
loads are inversely proportional to the velocities of dis- 
placement ; and in this way the idea arose that the 
product of a body ("corpus," " moles ") into its velocity 
is the measure of force. This product Descartes re- 
garded as a constant quantity. Leibnijfe's opinion, 
however, is, that this meastire of force is only acci- j 
dentally the correct measure, in the case of the ma- 
chines. The true measure of force is different, and 
must be determined by the method which Galileo and 



THE EXTENSION OF THE PA'LYC/PLES. 275 

Huygens pursued. Every body rises by virtue of the Leibnitz on 

, ..... i-i 1 tne nieas- 

velocity acquired in its descent to a height exactly ureof force, 
equal to that from which it fell. If, therefore, we as- 
sume, that the same "force" is requisite to raise a 
body m a height 4/* as to raise a body \>n a height //, 
we must, since we know that in the first case the ve- 
locity acquired in descent is but twice as great as in 
the second, regard the product of a "body " into the 
square of its velocity as the measure of forte. 

In a subsequent treatise (1695), Leibnitz reverts to 
this subject. He here makes a distinction between 
simple pressure (vis mortua) and the force of a moving 
body vis viva), which latter is made up of the sum of 
the pressure- impulses. These impulses produce, in- 
deed, an "impetus" (mv), but the impetus produced 
is not the true measure of force ; this, since the cause 
must be equivalent to the effect, is (in conformity with 
the preceding considerations) determined by mv 2 . 
Leibnitz remarks further that the possibility of per- 
petual motion is excluded only by the acceptance of his 
measure of force. 

Leibnyfz, no more than Descartes, possessed a gen- The idea of 

r .. . ... mass in 

nine concept of mass. Where the necessity of such Leibnitz - 
an idea occurs, he speaks of a body (carpus'), of a load 
s), of different-sized bodies of the same specific 
gravity, and so forth. Only in the second treatise, and 
there only once, does the expression "massa " occur, 
in all probability borrowed from Newton. Still, to de- 
rive any definite results from Leibni/z's theory, we must 
associate with his expressions the notion of mass, as 
his successors actually did. As to the rest, Leibnitz's 
procedure is much more in accordance with the meth- 
ods of science than Descartes's. Two things, however, 
are confounded : the question of the measure of force 



276 



THE SCIENCE OF MECHANICS. 



In a sense, 
Descartes 
and Leib- 
nitz were 
each right. 



The dis- 
pute, the re- 
sult of mis- 
under- 
standings. 



The appli- 
cation of 
the funda- 
mental 
equations 
to variable 
forces. 



and the question of the constancy of the sums 2mv and 
2n/7' 2 . The two have in reality nothing to do with 
each other. With regard to the first question, we now 
know that both the Cartesian and the Leibnitzian meas- 
ure of force, or, rather, the measure of the effective- 
ness of a body in motion, have, each in a different 
sense, their justification. Neither measure, however, 
as Leibni/z himself correctly remarked, is to be con- 
founded with the common, Newtonian, measure of 
force. 

With regard to the second question, the later in- 
vestigations of Newton really proved that iox free ma- 
terial systems not acted on by external forces the Car- 
tesian sum 12 mv is a constant ; and the investigations 
of Huygens showed that also the sum 2/nv 2 is a con- 
stant, provided work performed by forces does not alter 
it. The dispute raised by Leibn/z rested, therefore, 
on various misunderstandings. It lasted fifty-seven 
years, till the appearance of D'Alembert's Traite de 
dynamiquc, in 1743. To the theological ideas of Des- 
cartes and Leibnitz, we shall revert in another place. 

5. The three equations above discussed, though 
they are only applicable to rectilinear motions produced 
by constant forces, may yet be considered \\\e funda- 
mental equations of mechanics. If the motion be recti- 
linear but the force variable, these equations pass by a 
slight, almost self-evident, modification into others, 
which we shall here only briefly indicate, since mathe- 
matical developments in the present treatise are wholly 
subsidiary. 

From the first equation we get for variable forces 

m v = \ p dt -\- C, where / is the variable force, dt the 
time-element of the action, I p dt the sum of all the 



THE EXTENSION OF THE PRIXCFPLES. 277 

products / . dt from the beginning to the end of the 
action, and C a constant quantity denoting the value 
of m v before the force begins to act. The second equa- 
tion passes in like manner into the forms s = I dt\ dt 

-\-Ct + D, with two so-called constants of integration. 
The third equation must be replaced by 



J>*+c 



Curvilinear motion may always be conceived as the 
product of the simultaneous combination of three rec- 
tilinear motions, best taken in three mutually perpen- 
dicular directions. Also for the components of the mo- 
tion of this very general case, the above-given equa- 
tions retain their significance. 

6. The mathematical processes of addition, sub-Theunitso! 

...... . mechanics. 

traction, and equating possess intelligible meaning only 
when applied to quantities of the same kind. We can- 
not add or equate masses and times, or masses and 
velocities, but only masses and masses, and so on. 
When, therefore, we have a mechanical equation, the 
question immediately presents itself whether the mem- 
bers of the equation are quantities of the same kind, 
that is, whether they can be measured by the same unit, 
or whether, as we usually say, the equation is homo- 
tis. The units of the quantities of mechanics will 
form, therefore, the next subject of our investigations. 
The choice of units, which are, as we know, quan- 
tities of the same kind as those they serve to measure, 
is in many cases arbitrary. Thus, an arbitrary mass is 
employed as the unit of mass, an arbitrary length is 
employed as the unit of length, an arbitrary time as the 
unit of time. The mass and the length employed as 
units can be preserved ; the time can be reproduced 



278 THE SCIENCE OF MECHANICS. 

Arbitrary by pendulum-experiments and astronomical observa- 

units, and . .... ... 

derived or tions. r$ut units like a unit of velocity, or a unit of 
'units. acceleration, cannot be preserved, and are much more 
difficult to reproduce. These quantities are conse- 
quently so connected with the arbitrary fundamental 
units, mass, length, and time, that they can be easily 
and at once derived from them. Units of this class 
are called derived or absolute units. This latter desig- 
nation is due to Gauss, who first derived the magnetic 
units from the mechanical, and thus created the possi- 
bility of a universal comparison of magnetic measure- 
ments. The name, therefore, is of historical origin. 
The de- As unit of velocity we might choose the velocity 

of velocity, with which, say, q units of length are travelled over in 
tion, and unit of time. But if we did this, we could not express 
the relation between the time /, the distance s, and the 
velocity v by the usual simple formula s=ivt, but 
should have to substitute for it s = q .vt. If, however, 
we define the unit of velocity as the velocity with 
which the unit of length is travelled over in unit of 
time, we may retain the form s = vt. Among the de- 
rived units the simplest possible relations are made 
to obtain. Thus, as the unit of area and the unit of vol- 
ume, the square and cube of the unit of length are al- 
ways employed. 

According to this, we assume then, that by unit ve- 
locity unit length is described in unit time, that by unit 
acceleration unit velocity is gained in unit time, that 
by unit force unit acceleration is imparted to unit mass, 
and so on. 

The derived units depend on the arbitrary funda- 
mental units ; they are functions of them. The func- 
tion which corresponds to a given derived unit is called 
its dimensions. The theory of dimensions was laid down 



THE EXTENSION OF THE PRINCIPLES. 279 

by Fourier, in 1822, in his Theory of Heat. Thus, if /The theory 

. , , . , -it- of dimen- 

denote a length, / a time, and m a mass, the dimen- sions. 
sions of a velocity, for instance, are /// or lt~ x . After 
this explanation, the following table will be readily un- 
derstood : 

NAMES SYMBOLS DIMENSIONS 

Velocity V // _1 

Acceleration cp /t~ 2 

Force / w// -2 

Momentum mv mlt~ x 

Impulse // w// -1 

Work ps //-> 



/;/ 7 



Vis viva = w/ 2 /~ 2 

Moment of inertia & ////- 

Statical moment D m/ 2 /~ 2 

This table shows at once that the above-discussed equa- 
tions are homogeneous, that is, contain only members of 
the same kind. Ever) r new expression in mechanics 
might be investigated in the same manner. 

7. The knowledge of the dimensions of a quantity The usefui- 
is also important for another reason. Namely, if the Theory o'f e 
value of a quantity is known for one setof fundamental si'o^s" 
units and we wish to pass to another set, the value of 
the quantity in the new units can be easily found from 
the dimensions. The dimensions of an acceleration, 
which has, say, the numerical value cp, are 1 1~ 2 . If 
we pass to a unit of length A times greater and to a' 
unit of time r times greater, then a number A times 
smaller must take the place of / in the expression //~ 2 , 
and a number r times smaller the place of /. The 
numerical value of the same acceleration referred to 
the new units will consequently be (r 2 /A) cp. If we 



280 THE SCIENCE OE MECHANICS. 

take the metre as our unit of length, and the second as 
our unit of time, the acceleration of a falling body for 
example is 9-81, or as it is customary to write it, in- 
dicating at once the dimensions and the fundamental 
measures : 9-81 (metre/second 2 ). If we pass now to 
the kilometre as our unit of length (A = 1000), and to 
the minute as our unit of time (r = 60), the value of the 
same acceleration of descent is (60 X 60/1000) 9 81, 
or 35-316 (kilometre/minute 2 ). 
The inter- [8. The following statement of the mechanical units 

national L 

Bureau of at present in use in the United States and Great Britain 

Weights . x 

and Meas- is substituted for the statement by Professor Mach of 

ures. _ J 

the units formerly in use on the continent of Europe. 
All the civilised governments have united in establish- 
ing an International Bureau of Weights and Measures 
in the Pavilion de Breteuil, in the Pare of St. Cloud, 
at Sevres, near Paris. In some countries, the stan- 
dards emanating from this office are exclusively legal ; 
in others, as the United States and Great Britain, they 
are optional in contracts, and are usual with physi- 
cists. These standards are a standard of length and a 
standard of mass (not weigh/.) 
The inter- The unit of length is the International Metre, which 

unit of is defined as the distance at the melting point of ice 
between the centres of two lines engraved upon the 
polished surface of a platiniridium bar, of a nearly 
X-shaped section, called the International Prototype 
Metre. Copies of this, called National Prototype Me- 
tres, are distributed to the different governments. The 
international metre is authoritatively declared to be 
identical with the former French metre, used until the 
adoption of the international standard ; and it is im- 
possible to ascertain axiy error in this statement, be- 



length. 



THE EXTENSION OF I HE FRIXCIPLES. 

cause of doubt as to the length of the old metre, 
owing partly to the imperfections of the standard, and 
partly to obstacles now intentionally put in the way of 
such ascertainment. The French metre was defined 
as the distance, at the melting-point of ice, between 
the ends of a platinum bar, called the metre des archives. 
It was against the law to touch the ends, which made 
it difficult to ascertain the distance between them. 
Nevertheless, there was a strong suspicion they had 
been dented. The metre des archives was intended to 
be one ten-millionth of a quadrant of a terrestrial 
meridian. In point of fact such a quadrant is, ac- 
cording to Clarke, 32814820 feet, which is 10002015 
metres. 

The international unit of mass is the kilogramme, The inter- 
national 
which is the mass of a certain cylinder of platiniridium unit of 

mass 

called the International Prototype Kilogramme. Each 
government has copies of it called National Prototype 
Kilogrammes. This mass was intended to be identical 
with the former French kilogramme, which was defined 
as the mass of a certain platinum cylinder called the 
kilogramme des archives. The platinum being somewhat 
spongy contained a variable amount of occluded gases, 
and had perhaps suffered some abrasion. The kilo- 
gramme is 1000 grammes ; and a gramme was intended 
to be the mass of a cubic centimetre of water at its 
temperature of maximum density, about 3-93 C. It 
is not known with a high degree of precision how nearly 
this is so, owing to the difficulty of the determination. 

The regular British unit of length is the Imperial The British 
Yard which is the distance at 62 F. between the cen- length, 
tres of two lines engraved on gold plugs inserted in a 
bronze bar usually kept walled up in the Houses of 
Parliament in Westminster. These lines are cut rela- 



282 THE SCIENCE OF MECHANICS. 

Conditions tively deep, and the burr is rubbed off and the surface 

of compari- 
son of the rendered mat, by rubbing with charcoal. The centre 

Imperial 

Yard with of such a line can easily be displaced by rubbing ; which 

other meas- . ' * * 

arm. is probably not true of the lines on the Prototype me- 

tres. The temperature is, by law, ascertained by a 
mercurial thermometer ; but it was not known, at the 
time of the construction of the standard, that such 
thermometers may give quite different readings, ac- 
cording to the mode of their manufacture. The quality 
of glass makes considerable difference, and the mode 
of determining the fixed points makes still more. The 
best way of marking these points is first to expose the 
thermometer for several hours to wet aqueous vapor at 
a known pressure, and mark on its stem the height of 
the column of mercury. The thermometer is then 
brought down to the temperature of melting ice, as 
rapidly as possible, and is immersed in pounded ice 
which is melting and from which the water is not 
allowed to drain off. The mercury being watched 
with a magnifying glass is seen to fall, to come to 
rest, and to commence to rise, owing to the lagging 
contraction of the glass. Its lowest point is marked 
on the stem. The interval between the two marks is 
then divided into equal degrees. When such a ther- 
mometer is used, it is kept at the temperature to be 
determined for as long a time as possible, and imme- 
diately after is cooled as rapidly as it is safe to cool it, 
and its zero is redetermined. Thermometers, so made 
and treated, will give very constant indications. But 
the thermometers made at the Kew observatory, which 
are used for determining the temperature of the yard, 
are otherwise constructed. Namely the melting-point 
is determined first and the boiling-point afterwards ; 
and the thermometers are exposed to both tempera- 



THE EXTENSION OF THE PRINCIPLES. 2S3 

tures for many hours. The point which upon such a Relative 
thermometer will appear as 62 will really be consider- the metre 
ably hotter (perhaps a third of a centigrade degree) 
than if its melting point were marked in the other way. 
If this circumstance is not attended to in making com- 
parisons, there is danger of getting the yard too short 
by perhaps one two-hundred-thousandth part. General 
Comstock finds the metre equal to 39*36985 inches. 
Several less trustworthy determinations give nearly the 
same value. This makes the inch 2 540014 centimetres. 

At the time the United States separated from Eng- The Ameri- 

can unit of 

land, no precise standard of length was legal*; and length, 
none has ever been established. We are, therefore, 
without any precise legal yard ; but the United States 
office of weights and measures, in the absence of any 
legal authorisation, refers standards to the British Im- 
perial Yard. 

The regular British unit of mass is the Pound, de- The British 
fined as the mass of a certain platinum weight, called bub. 
the Imperial Pound. This was intended to be so con- 
structed as to be equal to 7000 grains, each the 5760th 
part of a former Imperial Troy pound. This would be 
within 3 grains, perhaps closer, of the old avoirdupois 
pound. The British pound has been determined by 
Miller to be o- 4535926525 kilogramme ; that is the kilo- 
gramme is 2-204621249 pounds. 

At the time the United States separated from Great 
Britain, there were two incommensurable units of 
weight, the avoirdupois pound and the Troy pound. Con- 
gress has since established a standard Troy pound, 
which is kept in the Mint in Philadelphia. It was a 
copy of the old Imperial Troy pound which had been 
adopted in England after American independence. It 

* The so-called standard'of 1758 had not been legalised. 



284 THE SCIENCE OF MECHANICS. 

TheAmeri- is a hollow brass weight of unknown volume ; and no 

can unit of . ..... 111 

mass. accurate comparisons of it with modern standards have 

ever been published. Its mass is, therefore, unknown. 
The mint ought by law to use this as the standard of 
gold and silver. In fact, they use weights furnished 
by the office of weights and measures, and no doubt 
derived from the British unit; though the mint officers 
profess to compare these with the Troy pound of the 
United States, as well as they are able to do. The old 
avoirdupois pound, which is legal for most purposes, 
differed without much doubt quite appreciably from 
the British Imperial pound ; but as the Office of Weights 
and Measures has long been, without warrant of law, 
standardising pounds according to this latter, the legal 
1 avoirdupois pound has nearly disappeared from use of 
late years. The makers of weights could easily detect 
the change of practice of the Washington Office. 

Measures of capacity are not spoken of here, be- 
cause they are not used in mechanics. It may, how- 
ever, be well to mention that they are defined by the 
weight of water at a given temperature which they 
measure. 
The unit of The universal unit of time is the mean solar day or 
its one 86400th part, which is called a second. Side- 
real time is only employed by astronomers for special 
purposes. 

Whether the International or the British units are 
employed, there are two methods of measurement of 
mechanical quantities, the absolute and the gravitational. 
The absolute is so called because it is not relative to 
the acceleration of gravity at any station. This method 
was introduced by Gauss. 

The special absolute system, widely used by physi- 
cists in the United States and Great Britain, is called 



time. 



THE EXTENSION OF THE PRIXCIPLES. 285 

the Centimetre-Gramme-Second system. In this sys- The abso- 
tern, writing C for centimetre, G for gramme mass, of the 
and S for second, states and 

Great Brit- 



ain. 



the unit of length is C 

the unit of mass is G 

the unit of time is S 

the unit of velocity is C/S 

the unit of acceleration (which might 
be called a "galileo," because Gali- 
leo Galilei first measured an accele- 
ration) is C/S 2 

the unit of density is G/C 8 

the unit of momentum is G C S 

the unit of force (called a dyne) is . . . G C/S a 
the unit of pressure (called one mil- 
lionth of an absolute atmosphere) is . . G/CS 2 ; 
the unit of energy (</> viva, or work, 

called an erg) is GC 2 /S 2 ; 

etc. 

The gravitational system of measurement of me- The Gravi- 
chanical quantities, takes the kilogramme or pound, or system, 
rather the attraction of these towards the earth, com- 
pounded with the centrifugal force, which is the ac- 
celeration called gravity, and denoted by g, and is dif- 
ferent at different places, as the unit of force, and 
the foot-pound or kilogramme-metre, being the amount 
of gravitational energy transformed in the descent of a 
pound through a foot or of a kilogramme through a 
metre, as the unit of energy. Two ways of reconciling 
these convenient units with the adherence to the usual 
standard of length naturally suggest themselves, namely, 
first, to use the pound weight or the kilogramme weight 
divided by g as the unit of mass, and, second, to adopt 



286 



THE SCIENCE OF MECHANICS, 



Compari- 
son of the 
absolute 
and gravi- 
tational 
tyatema. 



such a unit of time as will make the acceleration of g, 
at an initial station, unity. Thus, at Washington, the 
acceleration of gravity is 980 05 galileos. If, then, 
we take the centimetre as the unit of length, and tliL- 
0-031943 second as the unit of time, the acceleration 
of gravity will be 1 centimetre for such unit of time 
squared. The latter system would be for most pur- 
poses the more convenient ; but the former is the more 
familiar. 

In either system, the formula f> = ///g is retained: 
but in the former g retains its absolute value, while in 
the latter it becomes unity for the initial' station. In 
Paris, g is 980-96 galileos ; in Washington it is 980-05 
galileos. Adopting the more familiar system, and 
taking Paris for the initial station, if the unit of force 
is a kilogramme's weight, the unit of length a centi- 
metre, and the unit of time a second, then the unit of 
mass will be 1/981-0 kilogramme, and the unit of 
energy will be a kilogramme-centimetre, or (1/2)- 
(1000/981 -o) G C 2 /S 2 . Then, at Washington the 
gravity of a kilogramme will be, not 1, as at Paris, 
but 980 1/98 1 -0 = 0-99907 units or Paris kilogramme- 
weights. Consequently, to produce a force of one Pans 
kilogramme-weight we must allow Washington gravity 
to act upon 981 -0/980-1 = 1 -00092 kilogrammes.]* 

In mechanics, as in some other branches of physics 
closely allied to it, our calculations involve but three 
fundamental quantities, quantities of space, quantities 
of time, and quantities of mass. This circumstance is 
a source of simplification and power in the science 
which should not be underestimated. 

* For some critical remarks on the preceding method of exposition, see 
Nature, in the issue for November 15, 1894. 



THE EXTEXSIOX OF THE PRIXCIPLES. 287 



in. 

THE LAWS OF THE CONSERVATION OF MOMENTUM, OF THE 

CONSERVATION OF THE CENTRE OF GRAVITY, AND 

OF THE CONSERVATION OF AREAS. 

i. Although Newton's principles are fully adequate Specialisa- 
tion of the 
to deal with any mechanical problem that may arise, mechanical 

it is yet convenient to contrive for cases more frequently 
occurring, particular rules, which will enable us to treat 
problems of this kind by routine forms and to dis- 
pense with the minute discussion of them. Newton 
and his successors developed several such principles. 
Our first subject will be Newton's doctrines concern- 
ing freely movable material systems. 

2. If two free masses m and m' are subjected in Mutual ac- 
the direction of their line of junction to the action of masses, 
forces that proceed from other masses, then, in the in- 
terval of time /, the velocities v, v' will be generated, 
and the equation (/-)-/)/ = /?-)- m'v' will subsist. 
This follows from the equations pt = mv and p'f = 
m'v'. The sum mv -f m'v is called the momentum of 
the system, and in its computation oppositely directed 
forces and velocities are regarded as having opposite 
signs. If, now, the masses m, m' in addition to being 
subjected to the action of the external forces/, /' are 
also acted upon by internal forces, that is by such as 
are mutually exerted by the masses on one another, these 
forces will, by Newton's third law, be equal and op- 
posite, q, q. The sum of the impressed impulses 
is, then, (/ -f- /' -f q q) I = (p -\- p') t, the same as 
before ; and, consequently, also, the total momentum 
of the system will be the same. The momentum of a 



288 THE SCIENCE OF MECHANICS. 

system is thus determined exclusively by external forces, 
that is, by forces which masses outside of the system 
exert on its parts. 
Law of the Imagine a number of free masses m, m'. m" . . . . 

Conserva- ..... 

tionof Mo- distributed in any manner in space and acted on by 



men tn in. 



external forces/,/',/". . . . whose lines have any di- 
rections. These forces produce in the masses in the 
interval of time / the velocities v, v', v". . . . Resolve 
all the forces in three directions x, y, z at right angles 
to each other, and do the same with the velocities. 
The sum of the impulses in the ^'-direction will be equal 
to the momentum generated in the ^--direction ; and 
so with the rest. If we imagine additionally in action 
between the masses ///, ///', m" . . . ., pairs of equal and 
opposite internal forces^, q, r, r, s, s, etc., 
these forces, resolved, will also give in every direction 
pairs of equal and opposite components, and will con- 
sequently have on the sum-total of the impulses no in- 
fluence. Once more the momentum is exclusively de- 
termined by external forces. The law which states 
this fact is called the law of the conservation of momen- 
tum. 
Law of the 3. Another form of the same principle, which New- 
tion of the ton likewise discovered, is called the law of the conser- 

Centre of ... 

Gravity. ration of the centre oj grav- 

2 in fn : . . . _ 

J; -5 ? 7? T l 0'- Imagine in A and b 

(Fig. 149) two masses, itn 
Flg ' I+9 ' and m, in mutual action, 

say that of electrical repulsion ; their centre of gravity 
is situated at S, where S= 2AS. The accelerations 
they impart to each other are oppositely directed and 
in the inverse proportion of the masses. If, then, in 
consequence of the mutual action, o.m describes a dis- 
tance AD, m will necessarily describe a distance BC = 



THE EXTEXSIOX OF THE PRIXCIPLES. 289 

1AD. The point .S" will still remain the position of the 
centre of gravity, as CS = iDS. Therefore, two masses 
cannot, by mutual action, displace their common centre 
of gravity. 

If our considerations involve several masses, dis- This law 
tributed in any way in space, the same result will also systems of 

masses. 

be found to hold good for this case. For as no two of 
the masses can displace their centre of gravity by mu- 
tual action, the centre of gravity of the system as a 
whole cannot be displaced by the mutual action of its 
parts. 

Imagine freely placed in space a system of masses 
///, ///, 7n" . . . . acted on by external forces of any kind. . 
We refer the forces to a system of rectangular coordi- 
nates and call the coordinates respectively x, y\ z, x', 

'. and so forth. The coordinates of the centre of 
gravity are then 

p 2rnx _2my 2mz 

* "~ ~2m~' V ~ ~2m' *~ ~2nT' 

in which expressions x, y, z may change either by uni- 
form motion or by uniform acceleration or by any other 
law, according as the mass in question is acted on by 
no external force, by a constant external force, or by a 
variable external force. The centre of gravity will have 
in all these cases a different motion, and in the first 
may even be at rest. If now internal forces, acting be- 
tween every two masses, m and in" , come into play in 
the system, opposite displacements w', vf' will thereby 
be produced in the direction of the lines of junction 
of the masses, such that, allowing for signs, m'w' -f- 
m"iv" = 0. Also with respect to the components x x 
and x 2 of these displacements the equation m'x x + 
m"x 2 = will hold. The internal forces consequently 



2QO THE SCIENCE OF MECHANICS. 

produce in- the expressions B, y, B, only such additions 
as mutually destroy each other. Consequently, the 
motion of the centre of gravity of a system is determined 
by external forces only. 
Acceiera- If we wish to know the acceleration of the centre of 

tion of the . 

centre of gravity of the system, the accelerations ot the system s 

gravity of a , . .. . _. in i 

system. parts must be similarly treated. If cp, cp , cp . . . . de- 
note the accelerations of m, m', pi". ... in any direc- 
tion, and cp the acceleration of the centre of gravity in 
the same direction, <p= 2mcp/2m, or putting the 
total mass 2 m = M, cp = 2m <p/M. Accordingly, we 
obtain the acceleration of the centre of gravity of a 
system in any direction by taking the sum of all the 
forces in that direction and dividing the result by the 
total mass. The centre of gravity of a system moves 
exactly as if all the masses and all the forces of the 
system were concentrated at that centre. Just as a 
single mass can acquire no acceleration without the 
action of some external force, so the centre of gravity 
of a system can acquire no acceleration without the 
action of external forces. 

4. A few examples may now be given in illustra- 
tion of the principle of the conservation of the centre 
of gravity. 
Movement Imagine an animal free in space. If the animal 
ma'Tireein move in one direction a portion m of its mass, the re- 
space, mainder of it M will be moved in the opposite direction, 
always so that its centre of gravity retains its original 
position. If the animal draw back the mass ;//, the 
motion of M also will be reversed. The animal is un- 
able, without external supports or forces, to move itself 
from the spot which it occupies, or to alter motions im- 
pressed upon it from without. 

A lightly running vehicle A is placed on rails and 



THE EXTENSION OF THE PKIXCIPLES. 291 

loaded with stones. A man stationed in the vehicle of a ve- 

. hide, from 

casts out the stones one alter another, in the same di- which 

_. i 1 , t 1 r ston es are 

rection. 1 he vehicle, supposing the friction to be suf- cast, 
ficiently slight, will at once be set in motion in the op- 
posite direction. The centre of gravity of the system 
as a whole (of the vehicle + the stones) will, so far as 
its motion is not destroyed by external obstacles, con- 
tinue to remain in its original spot. If the same man 
were to pick up the stones from without and place 
them in the vehicle, the vehicle in this case would also 
be set in motion ; but not to the same extent as before, 
as the following example will render evident. 

A projectile of mass m is thrown with a velocity v Motion of a 

. cannon and 

from a cannon of mass ^}f. In the reaction, AT also re- s projec- 
tile, 
ceives a velocity, /', such that, making allowance for 

the signs, MV-\- mv = 0. This explains the so-called 
recoil. The relation here is V ' = (m/M)v; or, for 
equal velocities of flight, the recoil is less according as 
the mass of the cannon is greater than the mass of the 
projectile. If the work done by the powder be expressed 
by A, the vires viva will be determined by the equation 
Ml" 1 /2 -{- ;? ,2 /2 = A ; and, the sum of the momenta 
being by the first-cited equation = 0, we readily obtain 
V=V / 2Am/M(M-t-m). Consequently, neglecting 
the mass of the exploded powder, the recoil vanishes 
when the mass of the projectile vanishes. If the mass 
m were not expelled from the cannon but sucked into 
it, the recoil would take place in the opposite direc- 
tion. But it would have no time to make itself visible 
since before any perceptible distance had been trav- 
ersed, m would have reached the bottom of the bore. 
As soon, however, as il/and /// are in rigid connection 
with each other, as soon, that is, as they are relatively 
at rest to each other, they must be absolutely at rest, 



292 THE SCIENCE OF MECHANICS. 

for the centre of gravity of the system as a whole has 
no motion. For the same reason no considerable mo- 
tion can take place when the stones in the preceding 
example are taken into the vehicle, because on the 
establishment of rigid connections between the vehicle 
and the stones the opposite momenta generated are 
destroyed. A cannon sucking in a projectile would 
experience a perceptible recoil only if the sucked in 
projectile could fly through it. 
osciiia- Imagine a locomotive freely suspended in the air, 

tions of the , ... . , ... 

body of a or, what will subserve the same purpose, at rest with 
insufficient friction on the rails. By the law of the 
conservation of the centre of gravity, as soon as the 
heavy masses of iron in connection with the piston- 
rods begin to oscillate, the body of the locomotive will 
be set in oscillation in a contrary direction a motioi 
which may greatly disturb its uniform progress. Te 
eliminate this oscillation, the motion of the masses of 
iron worked by the piston-rods must be so compensated 
for by the contrary motion of other masses that the 
centre of gravity of the system as a whole will remaii 
in one position. In this way no motion of the body of 
the locomotive will take place. This is done by affix- 
ing masses of iron to the driving-wheels. 
illustration ' The facts of this case' may be very prettily showi 
case. 6 ' by Page's electromotor (Fig. 150). When the iroi 
core in the bobbin AB is projected by the internal forces 
acting between bobbin and core to the right, the bod} 
of the motor, supposing it to rest on lightly movable 
wheels rr, will move to the left. But if to a spoke of 
the fly-wheel R we affix an appropriate balance-weight 
a, which always moves in the contrary direction to the 
iron core, the sideward movement of the body of the 
motor may be made totally to vanish. 



THE EXTEXSIOX OF THE PRIXCIPLES. 293 

Of the motion of the fragments of a bursting bomb a bursting 

bomb. 

we know nothing. But it is plain, by the law of the 
conservation of the centre of gravity, that, making al- 
lowance for the resistance of the air and the obstacles 
the individual parts may meet, the centre of gravity of 
the system will continue after the bursting to describe 
the parabolic path of its original projection. 

;. A law closely allied to the law of the centre of Law of the 

J J . Conserva- 

gravity, and similarly applicable to free systems, is the ^ on a f 
principle of the conservation of areas. Although Newton 




Fig. 150. 

had, so to say, this principle within his very grasp, it 
was nevertheless not enunciated until a long time after- 
wards by Euler, D'Arcy, and Daniel Bernoulli. 
Euler and Daniel Bernoulli discovered the law almost 
simultaneously (1746), on the occasion of treating a 
problem proposed by Euler concerning the motion of 
balls in rotatable tubes, being led to it by the consider- 
ation of the action and reaction of the balls and the 
tubes. D'Arcy (1747) started from Newton's investiga- 
tions, and generalised the law of sectors which the 
latter had employed to explain Kepler's laws. 



294 



THE SCIEXCE OF MECHANICS. 



Deduction 
of the law. 



Two masses m, m' (Fig. 151) are in mutual action. 
By virtue of this action the masses describe the dis- 
tances AB, CD in the direction of their line of junction. 
Allowing for the signs, then, m . AB -f ;//. CD = 0. 
Drawing radii vectores to the moving masses from any 

point O, and regarding 
the areas described in 
opposite senses by the 
radii as having opposite 
signs, we further obtain 
///. OAB + ///'. OCD = 0. 
Which is to say, if two 
masses mutually act on 
each other, and radii vec- 
tores be drawn to these 
masses from any point, 
the sum of the areas 
described by the radii 
multiplied by the respec- 
tive masses is = 0. If the masses are also acted on 
by external forces and as the effect of these the areas 
OAE and OCEare described, the joint action of the 
internal and external forces, during any very small 
period of time, will produce the areas OAG and OCH. 
But it follows from Varignon's theorem that 

mOAG+ m'OCH= m OAE + m OCE + 
mOAB + m'OCD = mOAE + m'OCE; 

in other words, the sum of (he products of (he areas so de- 
scribed in(o (he rcspeciive masses which compose a system 
is unaliered by (he action of iniernal forces. 

If we have several masses, the same thing may be 
asserted, for every two masses, of the projection on any 
given plane of the motion. If we draw radii from 




THE EXTEXSIOX OF THE PRINCIPLES. 295 

any point to the several masses, and project on any 
plane the areas the radii describe, the sum of the 
products of these areas into the respective masses will 
be independent of the action of internal forces. This 
is the law of the conservation of areas. 

If a single mass not acted on by forces is moving imerpreta- 

, ; , 1 , 1 1 b tionofthe 

uniformly forward in a straight line and we draw a law. 
radius vector to the mass from any point 0, the area 
described by the radius increases proportionally to the 
time. The same law holds for 2mf, in cases in which 
several masses not acted on by forces are moving, 
where we signify by the summation the algebraic sum 
of all the products of the areas (/") into the moving 
masses a sum which we shall hereafter briefly refer 
to as the sum of the mass-areas. If internal forces 
come into play between the masses of the system, this 
relation will remain unaltered. It will still subsist, 
also, if external forces be applied whose lines of action 
pass through the fixed point 0, as we know from the 
researches of Newton. 

If the mass be acted on by an external force, the 
area / described by its radius vector will increase in 
time by the law/= at 2 ji -f- bt -\- c, where a depends 
on the accelerative force, b on the initial velocity, and 
c on the initial position. The sum 2 mf increases by 
the same law, where several masses are acted upon by 
external accelerative forces, provided these may be re- 
garded as constant, which for sufficiently small inter- 
vals of time is always the case. The law of areas in 
this case states that the internal forces of the system 
have no influence on the increase of the sum of the mass- 
areas. 

A free rigid body may be regarded as a system 
whose parts are maintained in their relative positions 



296 THE SCIENCE OE MECHANICS. 

Uniform ro- by internal forces. The law of areas is applicable there- 

tation of a . 

free rigid fore to this case also. A simple instance is afforded 
by the uniform rotation of a rigid body about an axis 
passing through its centre of gravity. If we call m a 
portion of its mass, r the distance of the portion from 
the axis, and a its angular velocity, the sum of the 
mass-areas produced in unit of time will be 2 m 
(j-/2)ra = (a/2)2m r 2 , or, the product of the moment 
of inertia of the system into half its angular velocity. 
This product can be altered only by external forces. 

illustrative 6. A few examples may now be cited in illustration 

examples. r .< i 

of the law. 

If two rigid bodies K and K' are connected, and K 
is brought by the action of internal forces into rotatioi 
relatively to K ', immediately K' also will be set in ro- 
tation, in the opposite direction.. The rotation of 
generates a sum of mass-areas which, by the law, must 
be compensated for by the production of an equal, but 
opposite, sum by A". 
Opposite This is very prettily exhibited by the electromotoi 

th e a whee of Fig. 1 52. The fly-wheel of the motor is placed ii 
a n freeeiec- a horizontal plane, and the motor thus attached to 
ro-motor. vert j ca i ax j Sj on w hich it can freely turn: The w T ires 
conducting the current dip, in order to prevent their 
interference with the rotation, into two conaxial gutters 
of mercury fixed on the axis. The body of the motor 
(K') is tied by a thread to the stand supporting the 
axis and the current is turned on. As soon as the fly- 
wheel (A"), viewed from above, begins to rotate in the 
direction of the hands of a watch, the string is drawn 
taut and the body of the motor exhibits the tendency 
to rotate in the opposite direction a rotation which im- 
mediately takes place when the thread is burnt away. 
The motor is, with respect to rotation about its 



THE EXTENSION OF THE PRIXUPLES. 



297 



axis, a free system. The sum of the mass-areas gen- itsexpiana 

tion bv the 

erated, for the case of rest, is = 0. But the wheel of law. 
the motor being set in rotation by the action of the in- 
ternal electro-magnetic forces, a sum of mass- areas is 




produced which, as the total sum must remain = 0, is 
compensated for by the rotation in the opposite direc- 
tion of the body of the motor. If an index be attached 
to the body of the motor and kept in a fixed position 



298 THE SCIENCE OF MECHANICS. 

by an elastic spring, the rotation of the body of the 
motor cannot take place. Yet every acceleration of 
the wheel in the direction of the hands of a watch (pro- 
duced by a deeper immersion of the battery) causes 
the index to swerve in the opposite direction, and every 
retardation produces the contrary effect, 
a variation A beautiful but curious phenomenon presents itself 
phenome 1 " 6 when the current to the motor is interrupted. Wheel 
and motor continue at first their movements in oppo- 
site directions. But the effect of the friction of the 
axes soon becomes apparent and the parts gradually 
assume with respect to each other relative rest. The 
motion of the body of the motor is seen to diminish ; 
for a moment it ceases ; and, finally, when the state of 
relative rest is reached, it is reversed and assumes the 
direction of the original motion of the wheel. The 
whole motor now rotates in the direction the wheel did 
at the start. The explanation of the phenomenon is 
obvious. The motor is not a perfectly free system. It 
is impeded by the friction of the axes. In a perfectly 
free system the sum of the mass-areas, the moment 
the parts re-entered the state of relative rest, would 
again necessarily be = 0. But in the present instance, 
an external force is introduced the friction of the 
axes. The friction on the axis of the wheel diminishes 
the mass-areas generated by the wheel and body of 
the motor alike. But the friction on the axis of the 
body of the motor only diminishes the sum of the mass- 
areas generated by the body. The wheel retains, thus, 
an excess of mass-area, which when the parts are rela- 
tively at rest is rendered apparent in the motion of the 
entire motor. The phenomenon subsequent to the in- 
terruption of the current supplies us with a model of 
what according to the hypothesis of astronomers has 



THE EXTEXSIOX OF THE PKIXCIPLES. 299 

taken place on the moon. The tidal wave created by its uiustra- 

. . . . tion in the 

the earth has reduced to such an extent by friction the case of the 

moon. 

velocity of rotation of the moon that the lunar day has 
grown to a month. The fly-wheel represents the fluid 
mass moved by the tide. 

Another example of this law is furnished by reac- Reaction- 

wheels. 

tion-whcels. If air or gas be emitted from the wheel 
(Fig. 153**) in the direction of the short arrows, the 
whole wheel will be set in rotation in the direction of 
the large arrow. In Fig. 153^, another simple reac- 
tion-wheel is represented. A brass tube rr plugged at 
both ends and appropriately perforated, is placed on a 
second brass tube R, supplied with a thin steel pivot 
through which air can be blown ; the air escapes at 
the apertures O, (7. 

It might be supposed that sucking on the reaction- variation 

b vv ... ofthephe- 

wheels would produce the opposite motion to that re-nomenaof 

reaction- 
Suiting from blowing. Yet this does not usually take wheels. 

place, and the reason is obvious. The air that is 

sucked into the spokes of the wheel must take part 

immediately in the motion of the wheel, must enter 

the condition of relative rest with respect to the wheel ; 

and when the system is completely at rest, the sum of 

its mass-areas must be = 0. Generally, no perceptible 

rotation takes place on the sucking in of the air. The 

circumstances are similar to those of the recoil of a 

cannon which sucks in a projectile. If, therefore, an 

elastic ball, which has but one escape-tube, be attached 

to the reaction-wheel, in the manner represented in 

Fig. 153 a, and be alternately squeezed so that the 

same quantity of air is by turns blown out and sucked 

in, the wheel will continue rapidly to revolve in the 

same direction as it did in the case in which we blew 

into it. This is partly due to the fact that the air 



3 



THE SCIEXCE OE MECHANICS. 




Fig. 153 b. 



THE EXTEXS10X OF THE PRIXCIPLES. 



301 



) 



Fg- 154. 



sucked into the spokes must participate in the motion Expiana- 
of the latter and therefore can produce no reactional variations, 
rotation, but it also partly results from the difference 
of the motion which the air outside the tube assumes 
in the two cases. In blowing, the air flows out in jets, 
and performs rotations. In sucking, the air comes in 
from all sides, and has no distinct rotation. 

The correctness of this view is easily demonstrated. 
If we perforate the bottom of a hollow cylinder, a closed 
band-box for instance, and 
place the cylinder on the steel 
pivot of the tube R, after the 
side has been slit and bent in 
the manner indicated in Fig. 
154. the box will turn in the 
direction of the long arrow 
when blown into and in the 
direction of the short arrow when sucked on. The air, 
here, on entering the cylinder, can continue its rotation 
unimpeded, and this motion is accordingly compensated 
for by a rotation in the opposite direction. 

7. The following case also exhibits similar condi- Reaction- 
tions. Imagine a tube (Fig. 155a) which, running 
straight from a to b, turns at right angles 
to itself at the latter point, passes to c, 
describes the circle cdef, whose plane 
is at right angles to ab, and whose cen- 
tre is at b, then proceeds from / to g, 
and, finally, continuing the straight line 
ab, runs from g to //. The entire tube 
is free to turn on an axis ah. If we 
pour into this tube, in the manner in- 
dicated in Fig. 155 , a liquid, which flows in the di- 
rection cdef, the tube will immediately begin to turn 



tubes. 




Fig. 155 a. 



32 



THE SCIENCE OF MECHANICS. 



in the direction fedc. This impulse, however, ceases, 
the moment the liquid reaches the point/, and flowing 
out into the radius/^ is obliged to join in the motion 
of the latter. By the use of a constant stream of liquid, 

therefore, the rotation 
of the tube may soon 
be stopped. But if the 
stream be interrupted, 
the fluid, in flowing off 
through the radius fg, 
will impart to the tube 
a motional impulse in 
the direction of its own 
motion, cdef, and the 
tube will turn in this di- 
rection. All these phe- 
nomena are easily ex- 
plained by the law of 

areas. 4di.X)0U -*J 

The trade- winds, the ! 
deviation of the oceanic 
currents and of rivers, 
Foucault's pendulum 
experiment, and the 
like, may also be treated 
as examples of the law 
Additional of areas. Another pretty illustration is afforded by 
lions. bodies with variable moments of inertia. Let a body 
with the moment of inertia & rotate with the angular 
velocity a and, during the motion, let its moment 
of inertia be transformed by internal forces, say by 
springs, into"', a will then pass into a', where a = 
a'', that is a' = a {/&). On any considerable dimi- 
nution of the moment of inertia, a great increase of 




Fig. 155 b. 



THE EXTENSION OF THE rKIXCIPLES. 303 

angular velocity ensues. The principle might con- 
ceivably be employed, instead of Foucault's method, 
to demonstrate the rotation of the earth, [in fact, some 
attempts at this have been made, with no very marked 
success]. 

A phenomenon which substantially embodies the Rotating 

... . j-irii- i liquid in a 

conditions last suggested is the following. A glass funnel, 
funnel, with its axis placed in a vertical position, js_ 05 P^df^J 
rapidly filled with a liquid in such a manner that the ^y; 

stream does not enter in the direction of the axis but 
strikes the sides. A slow rotatory motion is thereby 
set up in the liquid which as long as the funnel is full, is 
not noticed. But when the fluid retreats into the neck 
of the funnel, its moment of inertia is so diminished 
and its angular velocity so increased that a violent 
eddy with considerable axial depression is created. 
Frequently the entire effluent jet is penetrated by an 
axial thread of air. 

8. If we carefully examine the principles of the Both pnn- 

. ciples are 

centre of gravity and of the areas, we shall discover in simply spe- 

. . cialcasesof 

both simply convenient the law of 

e J . JS2L Z&. action and 

modes of expression, for 
practical purposes, of 
a well-known property 
of mechanical phenom- 
ena. To the accelera- 
tion cp of one mass m Flg- 156# 
there always corresponds a contrary acceleration cp' of 
a second mass m', where allowing for the signs mep -j- 
' cp' = 0. To the force m cp corresponds the equal 
and opposite force m'qf. When any masses m and 
2 m describe with the contrary accelerations 2 cp and cp 
the distances iw and w (Fig. 156), the position of 
their centre of gravity S remains unchanged, and the 




304 THE SCIENCE OF MECHANICS. 

sum of their mass-areas with respect to any point 
is, allowing for the signs, 2 m.f-\-m. 2/= 0. This 
simple exposition shows us, that the principle of the 
centre of gravity expresses the same thing with respect 
to parallel coordinates that the principle of areas ex- 
presses with respect to polar coordinates. Both contain 
simply the fact of reaction. 
h.u they The principles in question admit of still another 

may also be . . . T .... 

construed simple construction. Just as a single body cannot, 

as generali- . . i-n r ir ~ < 

sations of without the influence of external forces, that is, without 
inertia. the aid of a second body, alter its uniform motion of 
progression or rotation, so also a system of bodies can- 
not, without the aid of a second system, on which it 
can, so to speak, brace and support itself, alter what 
may properly and briefly be called its mean velocity of 
progression or rotation. Both principles contain, thus, 
a generalised statement of the law of inertia, the correct- 
ness of which in the present form we not only see but 
feel. 
importance This feeling is not unscientific ; much less is it 
stinctive detrimental. Where it does not replace conceptual in- 
mechanicai sight but exists by the side of it, it is really the funda- 
mental requisite and sole evidence of a complete mastery 
of mechanical facts. We are ourselves a fragment of 
mechanics, and this fact profoundly modifies our mental 
life.* No one will convince us that the consideration 
of mechanico-physiological processes, and of the feel- 
ings and instincts here involved, must be excluded from 
scientific mechanics. If we know principles like those 
of the centre of gravity and of areas only in their ab- 
stract mathematical form, without having dealt with the 
palpable simple facts, which are at once their applica- 

* For the development of this view, see E. Mach, Grundlinien der Lehre 
von den Be-ajegungsempfindungen. (Leipsic : Engelmann, 1875.) 



THE EXTEXSIOX OF THE PRIXCIPLES. 305 

tion and their source, we only half comprehend them, 
and shall scarcely recognise actual phenomena as ex- 
amples of the theory. We are in a position like that 
of a person who is suddenly placed on a high tower 
but has not previously travelled in the district round 
about, and who therefore does not know how to inter- 
pret the objects he sees. 



THE LAWS OF IMPACT. 

i. The laws of impact were the occasion of the Historical 

1 , . .... position of 

enunciation of the most important principles of me- the Laws of 
chanics, and furnished also the first examples of the 
application of such principles. As early as 1639, a 
contemporary of Galileo, the Prague professor, Marcus 
1 (born in 1595), published in his treatise De Pro- 
portionc Mot us (Prague) a few results of his investiga- 
tions on impact. He knew that a body striking in 
elastic percussion another of the same size at rest, loses 
its own motion and communicates an equal quantity 
to the other. He also enunciates, though not always 
with the requisite precision, and frequently mingled 
with what is false, other propositions which still hold 
good. Marcus Marci was a remarkable man. He pos- 
sessed for his time very creditable conceptions regard- 
ing the composition of motions and "impulses." In 
the formation of these ideas he pursued a method sim- 
ilar to that which Roberval later emploj'ed. He speaks 
of partially equal and opposite motions, and of wholly 
opposite motions, gives parallelogram constructions, 
and the like, but is unable, although he speaks of an 
accelerated motion of descent, to reach perfect clear- 
ness with regard to the idea of force and cons equently 
also with regard to the composition of forces. jTn spite Suk$t)t 



3 o6 



THE SCIENCE OF MECHANICS. 



Marcus 8 garding the descent of bodies in 



Marci. 




relating to the motion 



of th 



also a few propositions rei*u* { {orce an 



so on. 



THE EXTEXSIOX OF THE PRINCIPLES. 



307 



year previously, we cannot, in view of the condition of 
things produced in Central Europe by the Thirty Years' 




War, assume that Marci was acquainted with them. 
Not only would the many errors in Marci's book thus 
be rendered unintelligible, but it would also have to 



308 THE SCIENCE OE MECHANICS. 

Thesourcesbe explained how Marci, as late as 1648, in a continu- 

1 ! Marci's ..... 

knowledge, ation of his treatise, could have found it necessary to 

defend the theorem of the chords of circles against the 

Jesuit Balthasar Conradus. An imperfect oral com- 

- ,- munication of Galileo's researches is the more reason- 

able conjecture.* / When we add to all this that Marci 

was on the very verge of anticipating Newton in the 
discovery of the composition of light, we shall recog 
nise in him a man of very considerable parts. His 
writings are a worthy and as yet but slightly noticed 
object of research for the historian of physics. Though 
Galileo, as the clearest-minded and most able of his 
contemporaries, boie away in this province the palm, 
we nevertheless see from Avritings of this class that he 
was not by any means alone in his thought and ways 
of thinking. 
There- 2. Galileo himself made several experimental at- 

searches of ... 

Galileo. tempts to ascertain the laws of impact ; but he was not 
in these endeavors wholly successful. He principally 
busied himself with the force of a body in motion, or 
with the ''force of percussion," as he expressed it, 
and endeavored to compare this force with the pressure 
of a weight at rest, hoping thus to measure it. To this 
end he instituted an extremely ingenious experiment, 
which we shall now describe. 

A vessel I (Fig. 157) in whose base is a plugged 
orifice, is filled with water, and a second vessel II is 
hung beneath it by strings ; the whole is fastened to 
the beam of an equilibrated balance. If the plug is 
removed from the orifice of vessel I, the fluid will fall 

* I have been convinced, since the publication of the. first edition of this 
work, (see E. Wohlwill's researches, Die Entdeckung des Beharrungsgesitwc, 
in the Zeitscliriftfu>- VSlkerpsychologie, 1884, XV, page 387,) that Marcus Marci 
derived his information concerning the motion of falling bodies, from Galileo's 
earlier Dialogues. He may also have known the works of Kenedetti. 



THE EXTEXSIOX OF THE PRIXCIPLES. 



39 



in a jet into vessel II. A portion of the pressure due Galileo's 

, experi- 

to the resting weight of the water in I is lost and re- mem. 
placed by an action of impact on vessel II. Galileo 
expected a depression of the whole scale, by which he 
hoped with the assistance of a counter-weight to de- 
termine the effect of the impact. He was to some ex- 
tent surprised to obtain no depression, and he was un- 
able, it appears, perfectly to clear up the matter in his 
mind. 

3. To-day, of course, the explanation is not diffi- 
cult. By the removal of the plug there is produced, 




FR- 157- 

first, a diminution of the pressure. This consists of Expiana- 

t j j , ' on * ^ e 

two factors : (1) The weight of the jet suspended mexperi- 

, . . merit. 

the air is lost ; and (2) A reaction-pressure upwards is 
exerted by the effluent jet on vessel I (which acts like 
a Segner's wheel). Then there is an increase of pres- 
sure (Factor 3) produced by the action of the jet on the 
bottom of vessel II. Before the first drop has reached 
the bottom of II, we have only to deal with a diminu- 
tion of pressure, which, when the apparatus is in full 
operation, is immediately compensated for. This initial 



310 THE SCIENCE OF MECHANICS. 

Determina- depression was, in fact, all that Galileo could observe. 

tion of the . 

mechanical Let us imagine the apparatus in operation, and denote 

factors in- ,..._. . 

voived. the height the fluid reaches in vessel I by h, the corre- 
sponding velocity of efflux by v, the distance of the 
bottom of I from the surface of the fluid in II by k, the 
velocity of the jet at this surface by w, the area of the 
basal orifice by a, the acceleration of gravity by g, and 
the specific gravity of the fluid by s. To determine 
Factor (i) we may observe that v is the velocity ac- 
quired in descent through the distance h. We have, 
then, simply to picture to ourselves this motion of de- 
scent continued through k. The time of descent of 
the jet from I to II is therefore the time of descent 
through h -f- k less the time of descent through //. 
During this time a cylinder of base a is discharged 
with the velocity v. Factor (i), or the weight of the 
jet suspended in the air, accordingly amounts to 



l/2W* 



^p-m 



To determine Factor (2) we employ the familiar 
equation mv = pt. If we put /= I, then mv p, that 
is the pressure of reaction upwards on I is equal to the 
momentum imparted to the fluid jet in unit of time. 
We will select here the unit of weight as our unit of 
force, that is, use gravitation measure. We obtain for 
Factor (2) the expression \av{s lg)\v = p, (where the 
expression in brackets denotes the mass which flows 
out in unit of time,) or 

aV / 2gJi . . V^gTi =2a/is. 
Similarly we find the pressure on II to be 
w = q, or factor 3 : 



THE EXTEXSIOX OF THE PRIXCIPLES. 311 

f Mathemat- 

a - I ^Igh V-lg(h + k). ^eve'- 

* the result. 

The total variation of the pressure is accordingly 



laAs 



as 



+ -y/2ghV2g{h + k) 

or, abridged, 

2as[V A(A-\- k) A] 2a h s 
-f 2aJl/>4(>4 + ^); 

which three factors completely destroy each other. In 
the very necessity of the case, therefore, Galileo could 
only have obtained a negative result. 

We must supply a brief comment respecting Fac- a comment 

suggested 

tor (2). It might be supposed that the pressure on the by the ex- 

pertinent. 

basal orifice which is lost, is a As and not 2a As. But 
this statical conception would be totally inadmissible 
in the present, dynamical case. The velocity v is not 
generated by gravity instantaneously in the effluent 
particles, but is the outcome of the mutual pressure 
between the particles flowing out and the particles left 
behind ; and pressure can only be determined by the 
momentum generated. The erroneous introduction of 
the value a As would at once betray itself by self-con- 
tradictions. 

If Galileo's mode of experimentation had been less 
elegant, he would have determined without much diffi- 
culty the pressure which a continuous fluid jet exerts. 
But he could never, as he soon became convinced, 
have counteracted by a pressure the effect of an instan- 
taneous impact. Take and this is the supposition of 



312 



THE SCIENCE OF MECHANICS. 



Galileo's 
reasoning. 



Compari- 
son of the 
ideas im- 
pact and 
pressure. 



Galileo a freely falling, heavy body. Its final veloc- 
ity, we know, increases proportionately to the time. 
The very smallest velocity requires a definite portion 
of time to be produced in (a principle which even Mari- 
otte contested). If we picture to ourselves a body 
moving vertically upwards with a definite velocity, the 
body will, according to the amount of this velocity, 
ascend a definite time, and consequently also a definite 
distance. The heaviest imaginable body impressed 
in the vertical upward direction with the smallest im- 
aginable velocity will ascend, be it only a little, in 
opposition to the force of gravity. If, therefore, a 
heavy body, be it ever so heavy, receive an instan- 
taneous upward impact from a body in motion, be the 
mass and velocity of that body ever so small, and such 
impact impart to the heavier body the smallest imagin- 
able velocity, that body will, nevertheless, yield and 
move somewhat in the upward direction. The slightest 
impact, therefore, is able to overcome the greatest pres- 
sure ; or, as Galileo says, the force of percussion com- 
pared with the force of pressure is infinitely great. This 
result, which is sometimes attributed to intellectual ob- 
scurity on Galileo's part, is, on the contrary, a bril- 
liant proof of his intellectual acumen. We should say 
to-day, that the force of percussion, the momentum, 
the impulse, the quantity of motion m v, is a quantity 
of different dimensions horn the pressure/. The dimen- 
sions of the former are mlt~ x , those of the latter 7/1 1 1~ 2 . 
In reality, therefore, pressure is related to momentum 
of impact as a line is to a surface. Pressure is/, the 
momentum of impact is/ /. Without employing mathe- 
matical terminology it is hardly possible to express the 
fact better than Galileo did. We now also see why it 
is possible to measure the impact of a continuous fluid 



THE EXTEXSIOX OF THE PRINCIPLES. 313 

jet by a pressure. We compare the momentum de- 
stroyed per second of time with the pressure acting 
per second of time, that is, homogeneous quantities of 
the form //. 

4. The first systematic treatment of the laws ofThesyste- 

ii-i rm 1 r 1 malic treat- 

impact was evoked in the year 1608 by a request of the mem of the 

laws of im- 

Royal Society of London. Three eminent physicists pact. 
Wallis (Nov. 26, 1668), Wren (Dec. 17, 1668), and 
Huygens (Jan. 4, 1669) complied with the invitation of 
the society, and communicated to it papers in which, 
independently of each other, they stated, without de- 
ductions, the laws of impact. Wallis treated only of 
the impact of inelastic bodies, Wren and Huygens only 
of the impact of elastic bodies. Wren, previously to 
publication, had tested by experiments his theorems, 
which, in the main, agreed with those of Huygens. 
These are the experiments to which Newton refers in 
the Principia. The same experiments were, soon after 
this, also described, in a more developed form, by Ma- 
riotte, in a special treatise, Sur le Choc des Corps. Ma- 
riotte also gave the apparatus now known in physical 
collections as the percussion-machine. 

According to Wallis, the decisive factor in impact waiiis'sre- 

suits. 

is momentum, or the product of the mass (pondus) into 
the velocity (celerifas). By this momentum the force 
of percussion is determined. If two inelastic bodies 
which have equal momenta strike each other, rest will 
ensue after impact. If their momenta are unequal, 
the difference of the momenta will be the momentum 
after impact. If we divide this momentum by the sum 
of the masses, we shall obtain the velocity of the mo- 
tion after the impact. Wallis subsequently presented 
his theory of impact in another treatise, Mechanica sivc 
de Mot u. London, 1671. All his theorems may be 



3M 



THE SCIENCE OF MECHANICS. 



brought together in the formula now in common use, 
u = (inv -\- m'v')/(^m -\- m'), in which m, m denote the 
masses, v, v' the velocities before impact, and u the 
velocity after impact. 
Huygens's 5. The ideas which led Huygens to his results, are 
and results, to be found in a posthumous treatise of his, De Motu 
Corporum ex Percussione, 1703. We shall examine these 
in some detail. The assumptions from which Huygens 



O 

V 

Fig. 158. 






o o 



Fig. 159. 




An Illustration from De Percussione (Huygens). 

proceeds are : (1) the law of inertia ; (2) that elastic 
bodies of equal mass, colliding with equal and oppo- 
site velocities, separate after impact with the same ve- 
locities ; (3) that all velocities are relatively estimated ; 
(4) that a larger body striking a smaller one at rest 
imparts to the latter velocity, and loses a part of its 
own ; and finally (5) that when one of the colliding 
bodies preserves its velocity, this also is the case with 
the other. 



THE EXTEXSIOX OF THE PRLXCIPLES. 315 

Huvgens, now, imagines two equal elastic masses, First, equal 
which meet with equal and opposite velocities v. After masses ex- 
the impact they rebound from each other with exactly locitiec 
the same velocities. Huygens is right in assuming and 
not deducing this. That elastic bodies exist which re- 
cover their form after impact, that in such a transac- 
tion no perceptible vis viva is lost, are facts which ex- 
perience alone can teach us. Huygens, now, conceives 
the occurrence just described, to take place on a boat 
which is moving with the velocity v. For the specta- 
tor in the boat the previous case still subsists ; but for 
the spectator on the shore the velocities of the spheres 
before impact are respectively 2 v and 0, and after im- 
pact and 2 v. An elastic body, therefore, impinging 
on another of equal mass at rest, communicates to the 
latter its entire velocity and remains after the impact 
itself at rest. If we suppose the boat affected with any 
imaginable velocity, u, then for the spectator on the 
shore the velocities before impact will be respectively 
u -\- v and // v, and after impact // v and u -j- v. 
But since // -j- v and u v may have any values what- 
soever, it may be asserted as a principle that equal 
elastic masses exchange in impact their velocities. 

A bodv at rest, however great, is set in motion Second, the 

, , . .. . relative ve- 

bv a body which strikes it, however small; as Ga- locityofap- 

. proach and 

iileo pointed out. Huygens, now, recession is 

shows, that the approach of the ^K 

bodies before impact and their \?+ \ ) 

recession after impact take place * 

with the same relative velocity. A Flg ' l6a 

body m impinges on a body of mass M at rest, to which 
it imparts in impact the velocity, as yet undetermined, 
W. Huygens, in the demonstration of this proposition, 
supposes that the event takes place on a boat moving 



316 THE SCIENCE OE MECHANICS. 

from M towards m with the velocity w/2. The initial 
velocities are, then, v w/2 and w ji ; and the final 
velocities, x and -f- w / 2 - But as M has not altered 
the value, but only the sign, of its velocity, so ;//, if a 
loss of vis viva is not to be sustained in elastic impact, 
can only alter the sign of its velocity. Hence, the final 
velocities are (v w/2) and -f- w/2. As a fact, 
then, the relative velocity of approach before impact 
is equal to the relative velocity of separation after im- 
pact. Whatever change of velocity a body may suffer, 
in every case, we can, by the fiction of a boat in mo- 
tion, and apart from the algebraical signs, keep the 
value of the velocity the same before and after impact. 
The proposition holds, therefore, generally. 
Third,ifthe If two masses M and /// collide, with velocities V 
of approach and v inversely proportional to the masses, M after im- 
iy propor- pact will rebound with the velocity V and m with the 
masses, so velocity v. Let us suppose that the velocities after 
locitiesof impact are V x and z\ ; then by the preceding proposi- 
tion we must have V -\- v = V x -f- v x , and by the prin- 
ciple of vis viva 

MF 2 mv 2 _MV X 2 mv x 2 
~~2 ' 2~~ 2 ' 2 " 
Let us assume, now, that v t =v-{-w, then, neces- 
sarily, V x = V w ; but on this supposition 

MV 2 , mv 2 MV 2 , mv 2 , ,,> N w 2 

And this equality can, in the conditions of the case, 
only subsist if w = ; wherewith the proposition above 
stated is established. 

Huygens demonstrates this by a comparison, con- 
structively reached, of the possible heights of ascent 
of the bodies prior and subsequently to impact. If 



THE EXTEXSIOX OF THE PREXCIPLES. 317 

the velocities of the impinging bodies are not inversely This propo- 

sition. bv 

proportional to the masses, they may be made such by the fiction 

. ~, . . , of a moving 

the fiction of a boat in motion. The proposition thus boat, made 

to apply to 

includes all imaginable cases. ail cases. 

The conservation of vis viva in impact is asserted 
by Huygens in one of his last theorems (11), which he 
subsequently also handed in to the London Society. 
But the principle is unmistakably at the foundation of 
the previous theorems. 

6. In taking up the study of any event or phenom- Typical 

r J . modes of 

enon A, we may acquire a knowledge of its component natural in- 
elements by approaching it from the point of view of a 
different phenomenon B, which we already know ; in 
which case our investigation of A will appear as the 
application of principles before familiar to us. Or, we 
may begin our investigation with A itself, and, as na- 
ture is throughout uniform, reach the same principles 
originally in the contemplation of A. The investiga- 
tion of the phenomena of impact was pursued simul- 
taneously with that of various other mechanical pro- 
cesses, and both modes of analysis were really pre- 
sented to the inquirer. 

To begin with, we mav convince ourselves that the impact in 
* the New- 

problems of impact can be disposed of by the New- tonian 

tonian principles, with the help of only a minimum of view. 
new experiences. The investigation of the laws of im- 
pact contributed, it is true, to the discovery of New- 
ton's laws, but the latter do not rest solely on this foun- 
dation. The requisite new experiences, not contained 
in the Newtonian principles, are simply the informa- 
tion that there are elastic and inelastic bodies. Inelastic 
bodies subjected to pressure alter their form without 
recovering it ; elastic bodies possess for all their forms 
definite systems of pressures, so that every alteration 



3i8 



THE SCIEXCE OE MECHANICS. 



First, in- 
elastic 
masses. 



Impact in 
an equiva- 
lent point 
of view. 



of form is associated with an alteration of pressure, and 
vice versa. Elastic bodies recover their form ; and the 
forces that induce the form-alterations of bodies do not 
come into play until the bodies are in contact. 

Let us consider two inelastic masses M and m mov- 
ing respectively with the velocities Fand v. If these 
masses come in contact while possessed of these un- 
equal velocities, internal form-altering forces will be 
set up in the system M, m. These forces do not alter 
the quantity of motion of the system, neither do they 
displace its centre of gravity. With the restitution of 
equal velocities, the form-alterations cease and in in- 
elastic bodies the forces which produce the alterations 
vanish. Calling the common velocity of motion after 
impact u, it follows that Mu -f- mu = MV -\- Mv, or 
u = (MV-\- mv)/(M-\- m), the rule of Wallis. 

Now let us assume that we are investigating the 
phenomena of impact without a previous knowledge of 
Newton's principles. We very soon discover, when 
we so proceed, that velocity is not the sole determina- 
tive factor of impact; still another physical quality is 
decisive weight, load, mass, pondus, moles, massa. The 
moment we have noted this fact, the simplest case is 
easily dealt with. If two bodies of equal weight or 
equal mass collide with equal and 
opposite velocities ; if, further, the 
bodies do not separate after impact 
but retain some common velocity, 
plainly the sole uniquely deter- 
mined velocity after the collision is the velocity 0. If, 
further, we make the observation that only the dif- 
ference of the velocities, that is only relative velocity, 
determines the phenomenon of impact, we shall, by 
imagining the environment to move, (which experience 



O 



o 



Fig. 161. 



THE EXTEXSIOX OF THE PRINCIPLES. 



319 



tells us has no influence on the occurrence,) also readily 
perceive additional cases. For equal inelastic masses 
with velocities v and or v and v the velocity after 
impact is v/2. or (v -(- r')/2. It stands to reason that 
we can pursue such a line of reflection only after ex- 
perience has informed us what the essential and de- 
cisive features of the phenomena are. 

If we pass to unequal masses, we must not only 
know from experience that mass generally is of conse- 
quence, but also in what manner its influence is effec- 
tive. If, for example, two bodies of masses 1 and 3 
with the velocities v and /^collide, we might reason 



The expe- 
riential 
conditions 
of this 
method. 



V 






\ 



Fig. 162. 



Fig. 163. 



thus. We cut out of the mass 3 the mass 1 (Fig. 162), 
and first make the masses 1 and 1 collide : the result- 
ant velocity is (v -f- F)/2. There are now left, to 
equalise the velocities (v-\-V)/2 and V, the masses 

1-1 
gives 



2 and 2, which applying the same principle 

v+ r 



7, + 3F V + ZV 



2 4 1 + 3 

Let us now consider, more generally, the masses 
m and ///', which we represent in Fig. 163 as suitably 
proportioned horizontal lines. These masses are af- 
fected with the velocities v and v , which we represent 
by ordinates erected on the mass-lines. Assuming that 



320 THE SCIENCE OF MECHANICS. 

its points of m <; m', we cut off from m a portion ///. The offsetting 

contact . 

with the of /// and m gives the mass im with the velocity (v 4- 

Newtonian. 

v')/2. The dotted line indicates this relation. We 
proceed similarly with the remainder ?ri m. We cut 
off from 2; a portion m' ///, and obtain the mass 
im (m ///) with the velocity (j> -\- v')/2 and the 
mass 2 (;//' ni) with the velocity \{v + v')/2 + v'~\/2. 
In this manner we may proceed till we have obtained 
for the whole mass ?n -\- m the same velocity u. The 
constructive method indicated in the figure shows very 
plainly that here the surface equation (ni -|- ?') = 
mv -)- m'v subsists. We readily perceive, however, 
that we cannot pursue this line of reasoning except the 
sum m v -f- m'v , that is the form of the influence of m 
and v, has through some experience or other been pre- 
viously suggested to us as the determinative and de- 
cisive factor. If we renounce the use of the Newtonian 
principles, then some other specific experiences con- 
cerning the import of ;;/ v which are equivalent to those 
principles, are indispensable. 
Second, the j. The impact of elastic masses may also be treated 

impact of . , _ _ . ... _. 

elastic by the Newtonian principles. The sole observation 
Newton's here required is, that a deformation of elastic bodies 
calls into play forces of restitution, which directly de- 
pend on the deformation. Furthermore, bodies pos- 
sess impenetrability ; that is to say, when bodies af- 
fected with unequal velocities meet in impact, forces 
which equalise these velocities are produced. If two 
elastic masses M, m with the velocities C, c collide, a 
deformation will be effected, and this deformation will 
not cease until the velocities of the two bodies are 
equalised. At this instant, inasmuch as only internal 
forces are involved and therefore the momentum and 



THE EXTEXSIOX OF THE PRIXCIPLES. 321 

the motion of the centre of gravity of the system re- 
main unchanged, the common equalised velocity will be 
_ MC+mc 
U ~~ M+m ' 
Consequently, up to this time, M's velocity has suf- 
fered a diminution C // ; and w's an increase u c. 
But elastic bodies being bodies that recover their 
forms, in perfectly elastic bodies the very same forces 
that produced the deformation, will, only in the in- 
verse orda^ again be brought into play, through the 
very same elements of time and space. Consequently, 
on the supposition that /// is overtaken by M, M will a 
second time sustain a diminution of velocity C u, and 
m will a second time receive an increase of velocity 
u e. Hence, we obtain for the velocities V, v after 
impact the expressions V = 2 // C and v = 2u c, or 

v _ MC+m(2cC) _ mc+ M(2Ce) 
M+m ~' v= M+m 

If in these formulae we put M = m. it will follow V 1 * 5 led ; 1 ?- 

r Uon by this 

that V=c and v= C ; or, if the impinging masses are ^ e of ail 

* the laws. 

equal, the velocities which they have will be inter- 
changed. Again, since in the particular case Mjm = 
c/C or MC -f- m c = also u = 0, it follows that 
2 C = C and v = 2 c = e; that is, 
the masses recede from each other in this case with the 
same velocities (only oppositely directed) with which 
they approached. The approach of any two masses 
M, m affected with the velocities C, c, estimated as 
positive when in the same direction, takes place with 
the velocity C c ; their separation with the velocity 
V v. But it follows at once from F=2 C, 
v~lu~c, that V v = (Cc)) that is, the rela- 
tive velocity of approach and recession is the same. 



view. 



322 THE SCIE.YCE OF MECHANICS. 

By the use of the expressions V=2u C and v = 
2u c, we also very readily find the two theorems 

MV -\- m v = MC -j- m c and 
MV 2 + mv 2 = MC 2 + vie 2 , 

which assert that the quantity of motion before and 
after impact, estimated in the same direction, is the 
same, and that also the vis viva of the system before 
and after impact is the same. We have reached, thus, 
by the use of the Newtonian principles, ^dl of Huy- 
gens's results. 
The impii- 8. If we consider the laws of impact from Huygens's 

cations of . r . ,.,,. n . , . . 

Huygens's point of view, the following reflections immediately 
claim our attention. The height of ascent which the 
centre of gravity of any system of masses can reach is 
given by its vis viva, ^2mv 2 . In every case in which 
work is done by forces, and in such cases the masses 
follow the forces, this sum is increased by an amount 
equal to the work done. On the other hand, in every 
case in which the system moves in opposition to forces, 
that is, when work, as we may say, is done upon the 
system, this sum is diminished by the amount of work 
done. As long, therefore, as the algebraical sum of 
the work done on the system and the work done by the 
system is not changed, whatever other alterations may 
take place, the sum^2m7> 2 also remains unchanged. 
Huygens now, observing that this first property of ma- 
terial systems, discovered by him in his investigations 
on the pendulum, also obtained in the case of impact, 
could not help remarking that also the sum of the 
vires vivce must be the same before and after im- 
pact. For in the mutually effected alteration of the 
forms of the colliding bodies the material system con- 
sidered has the same amount of work done op. it as, on 



THE EXTEXSIOX OF THE PRIXCIPLES. 323 

the reversal of the alteration, is done by it, provided al- 
ways the bodies develop forces wholly determined by 
the shapes they assume, and that they regain their 
original form by means of the same forces employed to 
effect its alteration. That the latter process takes 
place, definite experience alone can inform us. This law 
obtains, furthermore, only in the case of so-called per- 
fectly elastic bodies. 

Contemplated from this point of view, the majority The dedoc- 

of the Huygenian laws of impact follow at once. Equal laws of im- 
pact by the 
masses, which strike each other with equal but oppo- notion of 

... i-ii 1 - - r>i vtsvfva and 

site velocities, rebound with the same velocities. The work, 
velocities are uniquely determined only when they are 
equal, and they conform to the principle of vis viva 
only by being the same before and after impact. Fur- 
ther it is evident, that if one of the unequal masses in 
impact change only the sign and not the magnitude of 
its velocity, this must also be the case with the other. 
On this supposition, however, the relative velocity of 
separation after impact is the same as the velocity of 
approach before impact. Every imaginable case can 
be reduced to this one. Let c and c be the velocities 
of the mass /// before and after impact, and let them be 
of any value and have any sign. We imagine the whole 
system to receive a velocity // of such magnitude that 
c = ( -f- c") or u = (c <')/- It will be seen 
thus that it is always possible to discover a velocity of 
transportation for the system such that the velocity of 
one of the masses will only change its sign. And so 
the proposition concerning the velocities of approach 
and recession holds generally good. 

As Huygens's peculiar group of ideas was not fully 
perfected, he was compelled, in cases in which the ve- 
locity-ratios of the impinging masses were not origin- 



324 



THE SCIENCE OF MECHANICS. 



Huygens's 
tacit appro- 
priation of 
the idea of 
mass. 



Construc- 
tive com- 
parison of 
the special 
and general 
case of im- 
pact. 



ally known, to draw on the Galileo-Newtonian svtem 
for certain conceptions, as was pointed out above. 
Such an appropriation of the concepts mass and mo- 
mentum, is contained, although not explicitly ex- 
pressed, in the proposition according to which the ve- 
locity of each impinging mass simply changes its sign 
when before impact Mjm = c/C. If Huygens had 
wholly restricted himself to his own point of view, he 
would scarcely have discovered this proposition, al- 
though, once discovered, he was able, after his own 
fashion, to supply its deduction. Here, owing to the 
fact that the momenta produced are equal and oppo- 
site, the equalised velocity of the masses on the com- 
pletion of the change of form will be u = 0. When the 
alteration of form is reversed, and the same amount of 
work is performed that the system originally suffered, 
the same velocities with opposite signs will be restored. 
If we imagine the entire system affected with a ve- 
locity of translation, this particular case will simulta- 
neously present the general ca.se. 
Let the impinging masses be 
represented in the figure by 
M=BC and m = AC (Fig. 
164), and their respective velo- 
cities by C= AD and c = BE. 
On AB erect the perpendicular 
CF, and through F draw IK 
parallel to AB. Then ID = (m. (T^c)/(M+ m) and 
KE = (M . C c)/(M-\- m). On the supposition now 
that we make the masses M and tn collide with the 
velocities ID and KE, while we simultaneously impart 
to the system as a whole the velocity 

u = AI=KB= C (m . C^~c)/(M + mj = 
e+(M. C-^c)l{M+ ni) == (MC + mc)/(M+ m}, 




Fig. 164. 



THE EXTENSION OF THE PRINCIPLES. 



325 



the' spectator who is moving forwards with the velocity 
// will see the particular case presented, and the spec- 
itor who is at rest will see the general case, be the 
,-elocities what they may. The general formulae of im- 
>act, above deduced, follow at once from this concep- 
ion. We obtain : 

m(Cc) MC+m(2cC) 



AG= C'l 



v = BH=c + 2 



M+ m 

M(Cc) 

M -\- m 



M-j- m 
mc + M(2C- 







M + m 



Huygen's successful employment of the fictitious signifi- 

. . . canceofthe 

lotions is the outcome of the simple perception that fictitious 

rr i-i--j motions. 

)dies not affected with differences of velocities do not 
ict on one another in impact. All forces of impact are 
letermined by differences of velocity (as all thermal 
effects are determined by differences of temperature), 
md since forces generally determine, not velocities, 
)ut only changes of velocities, or, again, differences of 
.elocities, consequently, in every aspect of impact the 

)le decisive factor is differences of velocity. With re- 
spect to which bodies the velocities are estimated, is 
indifferent. In fact, many cases of impact which from 
lack of practice appear to us as different cases, turn 
)ut on close examination to be one and the same. 

Similarly, the capacity of a moving body for work, Velocity, a 
whether we measure it with respect to the time of its level, 
iction by its momentum or with respect to the distance 

irough which it acts by its vis viva, has no signific- 
ance referred to a single body. It is invested with 
such, only when a second body is introduced, and, in 

le first case, then, it is the difference of the veloci- 
ies, and in the second the square of the difference that 
is decisive. Velocity is a physical level, like tempera- 
ture, potential function, and the like. 



326 THE SCIENCE OE MECHANICS. 

Possible It remains to be remarked, that Huygens could 

different . . J 

origin of have reached, originally, in the investigation of the 

Huygens's . , 

ideas. phenomena of impact, the same results that he pre- 
viously reached by his investigations of the pendulum. 
In every case there is one thing and one thing only to 
be done, and that is, to discover in all the facts the same 
elements, or, if we will, to rediscover in one fact the 
elements of another which we already know. From 
which facts the investigation starts, is, however, a 
matter of historical accident. 

Conserva- 9. Let us close our examination of this part of the 

tionofmo- . , 

mentum in- subject with a few general remarks. The sum of the 

terpreted. ..,,.. . 

momenta of a system ot moving bodies is preserved in 
impact, both in the case of inelastic and elastic bodies. 
But this preservation does not take place precisely in 
the sense of Descartes. The momentum of a body is 
not diminished in proportion as that of another is in- 
creased ; a fact which Huygens was the first to note. 
If, for example, two equal inelastic masses, possessed 
of equal and opposite velocities, meet in impact, the 
two bodies lose in the Cartesian sense their entire mo- 
mentum. If, however, we reckon all velocities in a 
given direction as positive, and all in the opposite as 
negative, the sum of the momenta is preserved. Quan- 
tity of motion, conceived in this sense, is always pre- 
served. 

The vis viva of a system of inelastic masses is al- 
tered in impact ; that of a system of perfectly elastic 
masses is preserved. The diminution of vis viva pro- 
duced in the impact of inelastic masses, or produced 
generally when the impinging bodies move with a com- 
mon velocity, after impact, is easily determined. Let 
M, m be the masses, C, c their respective velocities be- 



THE EXTEXSIOX OE THE PRIXCIPLES. 327 

fore impact, and // their common velocity after impact ; Conserva- 

p tion of vis 

then the loss of vis viva is m im- 

pact inter- 
^J/C- + \mr- (M+ **)**, (l)preted. 

which in view of the fact that // = (AfC-\- mc)/(M-\- w) 
may be expressed in the form (Mm/Af-\- ///) (C c) 2 . 
Carnot has put this loss in the form 

-JJ/(C-) 2 + i"'("-0 3 ( 2 ) 

If we select the latter form, the expressions \M{C u) 2 
and \m{u c) 2 will be recognised as the vis viva gen- 
erated by the work of the internal forces. The loss of 
vis viva in impact is equivalent, therefore, to the work 
done by the internal or so-called molecular forces. If 
we equate the two expressions (1) and (2), remember- 
ing that {M -\- ni)u = AfC -f- mc > we shall obtain an 
identical equation. Carnot's expression is important 
for the estimation of losses due to the impact of parts 
of machines. 

In all the preceding expositions we have treated oblique 
the impinging masses as points which moved only in the 
direction of the lines joining them. This simplifica- 
tion is admissible when the centres of gravity and the 
point of contact of the impinging masses lie in one 
straight line, that is, in the case of so-called direct im- 
pact. The investigation of what is called oblique im- 
pact is somewhat more complicated, but presents no 
especial interest in point of principle. 

A question of a different character was treated by The centre 

... Tri-i 1 1 .of percus- 

\\ allis. If a body rotate about an axis and its motion sion. 
be suddenly checked by the retention of one of its 
points, the force of the percussion will vary with the 
position (the distance from the axis) of the point ar- 
rested. The point at which the intensity of the impact 
is greatest is called by Wallis the centre of percussion. 



328 



THE SCIENCE OF MECHANICS. 



The ballis- 
tic pendu- 
lum. 



If this point be checked, the axis will sustain no pres- 
sure. We have no occasion here to enter in detail 
into these investigations ; they were extended and de- 
veloped by Wallis's contemporaries and successors in 
many ways. 

10. We will now briefly examine, before concluding 
this section, an interesting application of the laws of 
impact ; namely, the determination of the velocities of 
projectiles by the ballistic pendulum. A mass M is sus- 
pended by a weightless and massless 
string (Fig. 165), so as to oscillate as a 
pendulum. While in the position of 
equilibrium it suddenly receives the hori- 
zontal velocity V. It ascends by virtue 
of this velocity to an altitude h = (/) 
(1 cos a) = V 2 /'lg, where /denotes the 
length of the pendulum, a the angle of 
elongation, and g the acceleration of 
gravity. As the relation T= nV "l/g subsists between 
the time of oscillation T and the quantities /, g, we 
easily obtain V = (gT/ 7t) l/2(l cos^), and by the 
use of a familiar trigonometrical formula, also 




V. 



2 . a 

rZ'sm -jr 
71* 2 



itsformuia. If now the velocity Vis produced by a projectile of 
the mass m which being hurled with a velocity v and 
sinking in M is arrested in its progress, so that whether 
the impact is elastic or inelastic, in any case the two 
masses acquire after impact the common velocity/^, it. 
follows that mv = (M -\- ni) V; or, if m be sufficiently 
small compared with M, also v = {M/m)V; whence 

finally 

2 M . 4* 



THE EXTEXSIOX OF THE PRIXCIPLES. 329 

If it is not permissible to regard the ballistic pen- a different 

. deduction. 

dulum as a simple pendulum, our reasoning, in con- 
formity with principles before employed, will take the 
following shape. The projectile m with the velocity v 
has the momentum tnv, which is diminished by the 
pressure/ due to impact in a very short interval of 
time r to mV. Here, then, m {v V) = pr, or, if V 
compared with v is very small, mv = pr. With Pon- 
celet, we reject the assumption of anything like in- 
stantaneous forces, which generate instanter velocities. 
There are no instantaneous forces. What has been 
called such are very great forces that produce per- 
ceptible velocities in very short intervals of time, but 
which in other respects do not differ from forces that 
act continuously. If the force active in impact cannot 
be regarded as constant during its entire period of ac- 
tion, we have only to put in the place of the expression 
pr the expression fpt/t. In other respects the reason- 
ing is the same. 

A force equal to that which destroys the momentum The vis 
of the projectile, acts in reaction on the pendulum. If work of the 

pendulum. 

we take the line of projection of the shot, and conse- 
quently also the line of the force, perpendicular to the 
axis of the pendulum and at the distance b from it, the 
moment of this force will be bp, the angular accelera- 
tion generated bp/2mr 2 , and the angular velocity pro- 
duced in time r 



9 



b . pr btnv 

2/nr 2 2mr 2 



The vis viva which the pendulum has at the end of 
time r is therefore 

^^ - 2mr 2 



33Q THE SCIENCE OF MECHANICS. 

The result, By virtue of this vis viva the pendulum performs 

tll6 S3.II1C 

the excursion a, and its weight Mg, {a being the dis- 
tance of the centre of gravity from the axis,) is lifted 
the distance a (1 cos a). The work performed here 
is Mga{\ cos a), which is equal to the above-men- 
tioned vis viva. Equating the two expressions we 
readily obtain 

v 2 Mga 2m r 2 (\ cos a) 
mo 
and remembering that the time of oscillation is 



2mr 2 
\ Mga 

and employing the trigonometrical reduction which 
was resorted to immediately above, also 

i M a _ . a 

v = gl . sin . 

n m b 2 

interpreta- This formula is in every respect similar to that ob- 

tion of the . , 

result. tained for the simple case. The observations requisite 
for the determination of v, are the mass of the pendu- 
lum and the mass of the projectile, the distances of 
the centre of gravity and point of percussion from the 
axis, and the time and extent of oscillation. The form- 
ula also clearly exhibits the dimensions of a .velocity. 
The expressions 2/7T and sin(o , /2) are simple num- 
bers, as are also M/m and alb, where both numerators 
and denominators are expressed in units of the same 
kind. But the factor gT has the dimensions ii~ 1 , and 
is consequently a velocity. The ballistic pendulum 
was invented by Robins and described by him at length 
in a treatise entitled New Principles of Gunnery, pub- 
lished in 1742. 



THE EXTENSION OF THE PRINCIPLES. 331 



d'alembert's principle. 

1. One of the most important principles for the History of 

the prin- 

rr.pid and convenient solution of the problems of me- cipie. 
chanics is the principle of D'Alembert. The researches 
concerning the centre of oscillation on which almost all 
prominent contemporaries and successors of Huygens 
had employed themselves, led directly to a series of 
simple observations which D'Alembert ultimately gen- 
eralised and embodied in the principle which goes by 
his name. We will first cast a glance at these prelim- 
inary performances. They were almost without excep- 
tion evoked by the desire to replace the deduction of 
Huygens, which did not appear sufficiently obvious, by 
one that was more convincing. Although this desire was 
founded, as we have already seen, on a miscompre- 
hension due to historical circumstances, we have, of 
course, no occasion to regret the new points of view 
which were thus reached. 

2. The first in importance of the founders of the James Ber- 
theory of the centre of oscillation, after Huygens, iscomribu- 

tions to the 

James Bernoulli, who sought as early as 1686 to ex- theory of 
plain the compound pendulum by the lever. He ar- of osciiia- 
rived, however, at results which not only were obscure 
but also were at variance with the conceptions of Huy- 
gens. The errors of Bernoulli were animadverted on 
by the Marquis de L'Hopital in the Journal de Rotter- 
darn, in 1690. The consideration of velocities acquired 
in infinitely small intervals of time in place of velocities 
acquired infinite times a consideration which the last- 
named mathematician suggested led to the removal 



332 THE SCIENCE OF MECHANICS. 

of the main difficulties that beset this problem ; and in 
1691, in die Acta Eruditoruvi, and, later, in 1703, in the 
Proceedings of the Paris Academy James Bernoulli cor- 
rected his error and presented his results in a final and 
complete form. We shall here reproduce the essential 
points of his nnaldeduction. 
james Ber- A horizontal, massless bar AB (Fig. 166) is free to 

noulli'sde- v J 

auction of rotate about A; and at the distances r, r' from A the 

the law of 

the com- masses vi, in' are attached. The accelerations with which 

pound pen- 

duiumfrom these masses as thus connected 

the pnnci- m'r' X Rf 

pie of the ur 1 f Yz^A will fall must be different from 

lever. ^^--^ ** 

^^ I the accelerations which they 

^"^ _. ,, would assume if their connec- 

Fig. 166. 

tions were severed and they fell 
freely. There will be one point and one only, at the 
distance x, as yet unknown, from A which will fall 
with the same acceleration as it would have if it were 
free, that is, with the acceleration g. This point is 
termed the centre of oscillation. 

If in and ni were to be attracted to the earth, not 
proportionally to their masses, but m so as to fall when 
free with the acceleration cp = gr/x and in with the 
acceleration cp' = gr'/x, that is to say, if the natural 
accelerations of the masses were proportional to their 
distances from A, these masses would not interfere with 
one another when connected. In reality, however, m 
sustains, in consequence of the connection, an upward 
component acceleration g cp, and vi receives in virtue 
of the same fact a downward component acceleration 
cp' g; that is to say, the former suffers an upward 
force of vi (g <p) = g(x r/x) vi and the latter a 
downward force of vi' {cp' g) = g (r' x/x) m' . 

Since, however, the masses exert what influence 
they have on each other solely through the medium of 



THE EXTENSION OF THE PRIXCIPLES. 333 

the lever by which they are joined, the upward force The lawof 

,, , ,' , , thedistri- 

upon the one and the downward force upon the other bution of 

the effects 

must satisfy the law of the lever. If m in conse- of theim- 
quence of its being connected with the lever is held forces, in 

t 1 1 j 1 James Ber- 

back by a force/ from the motion which it would take, nouiiisex- 

ample. 

if free, it will also exert the same force /on the lever- 
arm r by reaction. It is this reaction pull alone that 
can be transferred to m' and be balanced there by a 
pressure f=z if/r')/, and is therefore equivalent to the 
latter pressure. There subsists, therefore, agreeably 
to what has been above said, the relation g (r' x/x) 
in = rjr 1 . g (x r/x) #* or, (x r) mr = {/ x) m'r, 
from which we obtain x = {nir 2 -\- tri V 2 ) / '{tn r -f- //), 
exactly as Huygens found it. The generalisation of 
this reasoning, for any number of masses, which need 
not lie in a single straight line, is obvious. 

3. Johx Bernoulli (in 171 2) attacked in a different The prin- 
manner the problem of the centre of oscillation. His John Ber- 
performances are easiest consulted in his Collected lution of 
Works {Opera, Lausanne and Geneva, 1762, Vols. Ilofthecen- 
and IV). We shall examine in detail here the main lation. 
ideas of this physicist. Bernoulli reaches his goal by 
conceiving the masses and forces separated. 

First,- let us consider two simple pendulums of dif- The first 
ferent lengths /, /' whose bobs are affected with gravi- Bernoulli's 
tational accelerations proportional to the lengths of the 
pendulums, that is, let us put ///' =g/g'. As the time 
of oscillation of a pendulum is T== nV Ijg, it follows 
that the times of oscillation of these pendulums will be 
the same. Doubling the length of a pendulum, ac- 
cordingly, while at the same time doubling the accel- 
eration of gravity does not alter the period of oscilla- 
tion. 

Second, though we cannot directly alter the accel- 



334 THE SCIENCE OE MECHANICS. 

The second eration of gravity at any one spot on the earth, we 

step in John . r 

Bernoulli's can do what amounts virtually to this. Thus, imagine 

deduction. . 

a straight massless bar of length ia, free to rotate about 
its middle point; and attach to the one ex- 
tremity of it the mass m and to the other the 
mass m' . Then the total mass is m -f- vi at 
the distance a from the axis. But the force 
which acts on it is (/;/ ;;/) g, and the ac- 
celeration, consequently, {m m'/m -f- ;//' ) g. 
Fig. 167. Hence, to find the length of the simple pen- 
dulum, having the ordinary acceleration of 
gravity g, which is isochronous with the present pen- 
dulum of the length a, we put, employing the preced- 
ing theorem, 

/ g m 4- ?n 

= = } or / = a 

a vi m m tn 

g 

VI -f- VI 

The third Third, we imagine a simple pendulum of length 1 

step, or the . . . 

determina- with the mass vi at its extremity. The weight of m 

tion of the . , ..,.,. 

centre of produces, by the principle of the lever, the same ac- 
celeration as half this force at a distance 2 from the 
point of suspension. Half the mass m placed at the 
distance 2, therefore, would suffer by the action of the 
force impressed at 1 the same acceleration, and a fourth 
of the mass vi would suffer double the acceleration ; so 
that a simple pendulum of the length 2 having the orig- 
inal force at distance 1 from the point of suspension 
and one-fourth the original mass at its extremity would 
be isochronous with the original one. Generalising 
this reasoning, it is evident that we may transfer any 
force f acting on a compound pendulum at any dis- 
tance r, to the distance 1 by making its value rf, and 
any and every mass placed at the distance r to the 
distance 1 by making its value r 2 vi, without changing 



THE EXTENSION OF THE PRINCIPLES. 335 

the time of oscillation of the pendulum. If a force / 
act on a lever-arm a (Fig. 168) while at the distance r 
from the axis a mass m is attached, / will be equiva- 
lent to a force afjr impressed on 
/// and will impart to it the linear q, 
acceleration af/mr and the angu- ~ 

lar acceleration af/mr 2 . Hence, 
to find the angular acceleration 
of a compound pendulum, we 

divide the sum of the statical moments by the sum of 
the moments of inertia. 

Brook Taylor, an Englishman,* also developed The re- 
this idea, on substantially the same principles, but Brook Tay- 
quite independently of John Bernoulli. His solution, 
however, was not published until some time later, in 
1 71 5, in his work, Methodus Incrementorum. 

The above are the most important attempts to solve 
the problem of the centre of oscillation. We shall see 
that they contain the very same ideas that D'Alembert 
enunciated in a generalised form. 

4. On a system of points M, M', M" . . . . connected Motion of a 

system of 

with one another in any way,f the forces P, P ', P". . . . pointssub- 
. , . ject to con- 

are impressed. (Fig. 169.) These forces would 1m- straims. 

part to the free points of the system certain determinate 

motions. To the connected points, however, different 

motions are usually imparted motions which could 

be produced by the forces IV, W, W" . . . . These 

last are the motions which we shall study. 

^ Conceive the force P resolved into JFand V, the 

force P' into W and /", and the force P" into W" 

* Author of Taylor's theorem, and also of a remarkable work on perspec- 
/ rans. 

1 in precise technical language, they are subject to constraints, that is. 
forces regarded as infinite, which compel a certain relation between their 
motions. Trans. 



336 



THE SCIENCE OF MECHANICS. 



Statement 
of D'Alem- 
bert's prin- 
ciple. 



Various 
forms in 
which the 
principle 
may be ex- 
pressed. 




Fig. 169. 



and V", and soon. Since, owing to the connections, 
only the components W, W , IV". . . . are effective, 
therefore, the forces V, V, V" . . . . must be equilib- 
rated by the connections. We will call the forces P, P', 

P" the impressed forces, 

the forces JV, W, JV" , 

which produce the ac- 
tual motions, the effectmt 
forces, and the forces V, 
V , V" . . . . the forces 
gained and lost, or the 
equilibrated forces. We 
perceive, thus, that if we 
resolve the impressed forces into the effective forces 
and the equilibrated forces, the latter form a system 
b alance d by the connections. This is the principle of 
D'Alembert. We have allowed ourselves, in its expo- 
sition, only the unessential modification of putting 
forces for the momenta generated by the forces. In this 
form the principle was stated by D'Alembert in his 
Traite de dynamique, published in 1743. 

As the system V, V, V". ... is in e quilibrium, the 
principle of virtual displacements is applicable thereto. 
This gives a second form of D'Alembert's principle. 
A third form is obtained as follows : The forces P, P'. . . . 
are the resultants of the components IV, W . . . . and 
V, V. . . . If, therefore, we combine with the forces 

W, W and V, V . . . . the forces P, P' , 

equilibrium will obtain. The force-system P, W, V 
is in equilibrium. But the system V\s independently 
in equilibrium. Therefore, also the system P, J Vis 
in equilibrium, or, what is the same thing, the system 
P, Wis in equilibrium. Accordingly, if the effective 
forces with opposite signs be joined to the impressed 



THE EXTEXSIOX OF THE PRINCIPLES. 



337 



forces, the two, owing to the connections, will balance. 
The principle of virtual displacements may also be ap- 
plied to the system P, IV. This Lagrange did in his 
Mccaniquc analytique, 1788. 

The fact that equilibrium subsists between the sys- .^nequh-a- 

n 1 .1 rr- ..'... lent princi- 



tem /"and the system IF, may be expressed in still p*" em" 10 ' 

ployed by 
Hermann 
and Eulei. 



//' 



Fig. 170. 



another way. We may say that 

the syste m Wis equival ent to the _p 

system P. In this form Her- " 

Manx 1 Phoronomia, 17 16) and 

Euler {Comment. Acad. Pet r op., 

Old Series, Vol. VII, 1740) employed the principle. 

It is substantially not different from that of D'Alembert. 

5. We will now illustrate D'Alembert' s principle by 
one or two examples. 

On a massless wheel and axle with the radii R, r the illustration 
loads P and Q are hung, which are not in equilibrium, bert-sprin- 
We resolve the force P into (1) W 
(the force which would produce the 
actual motion of the mass if this were 
free) and (2) V, that is, we put 
P W+ Fand also Q = W + V; 
it being evident that we may here 
disregard all motions that are not 
in the vertical. We have, accord- 
ingly, V=P JFand V=Q W, 
and, since the forces V, V are in equilibrium, also 
Y' R J = V'. r. Substituting for V, V in the last equa- 
tion their values in the former, we get 




the 
motion of a 
wheel and 
axle. 



W")r 



(1) 



which may also be directly obtained by the employ- 
ment of the second form of D'Alembert's principle. 
From the conditions of the problem we readily perceive 



338 THE SCIENCE OF MECHANICS. 

that we have here to deal with a uniformly accelerated 
motion, and that all that is therefore necessary is to 
ascertain the acceleration. Adopting gravitation meas- 
ure, we have the forces W and W, which produce in 
the masses Pjg and Q/g the accelerations y and y'\ 
wherefore, W=(P/g)y and IV' = (Q/g)y'. But we 
also know that y'= y{r/R). Accordingly, equation 
(i) passes into the form 

?-ty*={ Q+ iTr) r (2) 

whence the values of the two accelerations are ob- 
tained 

PR^Qr .,' PRQr 

v = plv + Q^ g ' r = ~ PJR^FQ^ rg ' 

These last determine the motion. 
Employ- It will be seen at a glance that the same result can 

ideas stat- be obtained by the employment of the ideas of statical 

ical mo- , ... _ TT ... . 

mem and moment and moment of inertia. We get by this metnoa 

moment of , . 

inertia, to for the angular acceleration 

PRQr PR Qr J 

and as y = R cp and y'= rep we re-obtain the pre- 
ceding expressions. 

When the masses and forces are given, the problem 
of finding the motion of a system is determinate. Sup- 
pose, however, only the acceleration y is given with 
which P moves, and that the problem is to find the loads 
P and Q that produce this acceleration. We obtain 
easily from equation (2) the result P= Q{Rg-{- ry) 
r/(g y)R 2 , that is, a relation between P and Q. 
One of the two loads therefore is arbitrary. The prob- 



obtain this 
result. 



THE EXTEXSIOX OF THE PRINCIPLES. 



339 



lem in this form is an indeterminate one, and may be 
solved in an infinite number of different ways. 

The following may serve as a second example. 

A weight P (Fig. 172) free to move on a 
straight line AB, is attached to a cord 
passing over a pulley and canning a 
weight Q at the other end. The cord 
makes with the line AB the variable 
angle a. The motion of the present 
case cannot be uniformly accelerated. 
But if we consider only vertical mo- 
tions we can easily give for every 
value of a the momentary' accelera- 
tion (y and y") of P and Q. Proceeding exactly as 
we did in the last case, we obtain 
p = w+ V, 
Qz=lV + V 



Vertical A second Vt 
lustration 
of the Prin- 
ze ciple. 




Fig. 172. 



also 



:osar = V, or, since y' = ycosa, 



Q 

Q -\- - cos a y 



cos a 



g 



whence 



P Q cos a 



g 



' Qcos 2 a-\-P' 

P Qcosa 

' Qcos 2 a -\- P 6 

Again the same result may be easily reached by the solution of 
employment of the ideas of statical moment and mo- also by^e 
ment of inertia in a more generalised form. The fol- stadcai mo- 
lowing reflexion will render this clear. The force, or momenTof 
statical moment, that acts on P is P Q cos a. But eiised Ren " 
the weight Q moves cosnr times as fast as P; conse- 
quently its mass is to be taken cos 2 a times. The ac- 
celeration which P receives, accordingly is, 



34Q THE SCIENCE OF MECHANICS. 
P QCOSOC P ^COSff 



r = 



Q , P Qcos 2 a + P 

cos 2 a -\ 



In like manner the corresponding expression for y' may 
be found. 

The foregoing procedure rests on the simple re- 
mark, that not the circular path of the motion of the 
masses is of consequence, but only the relative veloci- 
ties or relative displacements. This extension of the 
concept moment of inertia may often be employed to 
advantage, 
import and 6. Now that the application of D'Alembert's prin- 

character , , . j-rr 

of D'Aiem- ciple has been sufficiently illustrated, it will not be diffi- 

bert's prin- ...._ _.. 

ciple. cult to obtain a clear idea of its significance. Problems 

relating to the motion of connected points are here dis- 
posed of by recourse to experiences concerning the 
mutual actions of connected bodies reached in the in- 
vestigation of problems of equilibrimn. Where the last 
mentioned experiences do not suffice, D'Alembert's 
principle also can accomplish nothing, as the examples 
adduced will amply indicate. We should, therefore, 
carefully avoid the notion that D'Alembert's principle 
is a general one which renders special experiences su- 
perfluous. Its conciseness and apparent simplicity are 
wholly due to the fact that it refers us to experiences 
already in our possession. Detailed knowledge of the 
subject under consideration founded on exact and mi- 
nute experience, cannot be dispensed with. This knowl- 
edge we must obtain either from the case presented, 
by a direct investigation, or we must previously have 
obtained it, in the investigation of some other subject, 
and carry itwith us to the problem in hand. We learn, 
in fact, from D'Alembert's principle, as our examples 
show, nothing that we could not also have learned by 



THE EXTEXSIOX OF THE PRIXCIPLES. 341 

other methods. The principle fulfils in the solution 
of problems, the office of a routine-form which, to a 
certain extent, spares us the trouble of thinking out 
each new case, by supplying directions for the employ- 
ment of experiences before known and familiar to us. 
The principle does not so much promote our insight 
into the processes as it secures us a practical mastery of 
them. The value of the principle is of an economical 
character. 

When we have solved a problem by D'Alembert's The reia- 

-riii tion ot 

principle, we may rest satisfied with the experiences DAiem- 

previouslv made concerning equilibrium, the applica- cipie to the 

. . . .... ., . other prin- 

tion of which the principle implies. But if we wishcipiesof 

mechanics. 

clearly and thoroughly to apprehend the phenomenon, 
that is, to rediscover in it the simplest mechanical ele- 
ments with which we are familiar, we are obliged to 
push our researches further, and to replace our expe- 
riences concerning equilibrium either by the Newtonian 
or by the Huygenian conceptions, in some way similar 
to that pursued on page 266. If we adopt the former 
alternative, we shall mentally see the accelerated mo- 
tions enacted which the mutual action of bodies on one 
another produces ; if we adopt the second, we shall di- 
rectly contemplate the work done, on which, in the v .^- 
Huygenian conception, the r/x r/rvf depends. The latter 
point of view is particularly convenient if we employ 
the principle of virtual displacements to express the 
conditions of equilibrium of the system V or P IV. 
D'Alembert's principle then asserts, that the sum of 
the virtual moments of the system V, or of the system 
P IV, is equal to zero. The elementary work c r ihe 
equilibrated forces, if we leave out of account the strain- 
ing of the connections, is equal to zero. The total 
work done, then, is performed solely by the system P, 



342 THE SCIENCE OF MECHANICS. 

and the work performed by the system Fmust, accord- 
ingly, be equal to the work done by the system P. All 
the work that can possibly be done is due, neglecting 
the strains of the connections, to the impressed forces. 
As will be seen, D'Alembert's principle in this form is 
not essentially different from the principle of vis viva. 
Form of ap- 7. In practical applications of the principle of 

plication of ... . y r 

D'Aiem- D Alembert it is convenient to resolve every force P 

bert's prin- . 

cipie, and impressed on a mass ;;/ of the system into the mutuallv 

the result- " * 

ingequa- perpendicular components X, Y, Z parallel to the axes 

tionsofmo- , 

tion. of a system of rectangular coordinates ; every effective 

force W'm.\.o corresponding components mB,, mrj, mS,, 
where B,, t], 2, denote accelerations in the directions of 
the coordinates ; and every displacement, in a similar 
manner, into three displacements 6x, Sy, 8z. As the 
work done by each component force is effective only in 
displacements parallel to the directions in which the 
components act, the equilibrium of the system (P, W") 
is given by the equation 

2 \ (X m ) &c -f ( Y >mj) o> + (Z mg) 6z\z=0 (1) 

or 

2(X6x^Y&y + Z6n)=:2M(6x+i?ty + 26i). . (2) 

These two equations are the direct expression of the 
proposition above enunciated respecting the possible 
work of the impressed forces. If this work be = 0, the 
particular case of equilibrium results. The principle 
of virtual displacements flows as a special case from 
this expression of D'Alembert's principle ; and this is 
quite in conformity with reason, since in the general 
as well as in the particular case the experimental per- 
ception of the import of work is the sole thing of con- 
sequence. 

Equation (i) gives the requisite equations of mo- 



THE. EXTEXSIOX OF THE PRIXCIPLES. 343 

tion ; we have simply to express as many as possible 
of the displacements o\v, 6y, 6z by the others in terms 
of their relations to the latter, and put the coefficients 
of the remaining arbitrary displacements = 0, as was 
illustrated in our applications of the principle of vir- 
tual displacements. 

The solution of a very few problems by D'Alem- com-e- 
bert's principle will suffice to impress us with a full utility of 

i . . T ... , . D'Alem- 

sense ot its convenience. It will also give us the con- berfsprin- 
viction that it is possible, in every case in which it may 
be found necessary, to solve directly and with perfect 
insight the very same problem by a consideration of 
elementary mechanical processes, and to arrive thereby 
at exactly the same results. Our conviction of the 
feasibility of this operation renders the performance of 
it, in cases in which purely practical ends are in view, 
unnecessary. 



THE PRINXIPLE OF VIS VIVA . 

t. The principle of vis viva, as we know, was first The oriu- 
employed by Huygens. John and Daniel Bernoulli leaf fonnof 
had simply to provide for a greater generality of ex- cipie."" 
pression ; they added little. If /, /', /". . . . are weights, 
m, tri , m" . . . . their respective masses, h, K ', h" . . . . the 
distances of descent of the free or connected masses, 
and v, v'. v". . . . the velocities acquired, the relation 
obtains 

2 ph = |2/ r 2 . 

If the initial velocities are not =0, but are v .. 
. ., the theorem will refer to the increment of the 
Vis viva by the work and read 



344 



THE SCIENCE OF MECHANICS. 



Theprinci- The principle still remains applicable when i> . . . . 

pie applied . , 

tc forces of are, not weights, but any constant forces, and h . . . 

any kind. . . 

not the vertical spaces fallen through, but any paths in 
the lines of the forces. If the forces considered are 
variable, the expressions///, p'ti. . . . must be replaced 
by the expressions Cp ds, Cp' ds' . . . ., in which p de- 
notes the variable forces and ds the elements of dis- 
tance described in the lines of the forces. Then 

= $2m(v* v 2 ) 



Jpds+Jp'ds' + . 



or 



The princi- 
ple illus- 
trated by 
the motion 
of a wheel 
and axle. 



2fp ds = ^2 m (v 2 v 2 ) 



(1) 



2. In illustration of the principle of vis viva we 
shall first consider the simple problem which we treated 
by the principle of D'Alembert. On 
a wheel and axle with the radii R, r 
hang the weights P, Q. When this 
machine is set in motion, work is per- 
formed by which the acquired vis viva 
is fully determined. For a rotation of 
the machine through the angle a, the 
work is 

P. Ret Q . ra = a(PRQr). 
Calling the angular velocity which 
corresponds to this angle of rotation, cp, the vis viva 
generated will be 




F''g. 173. 



p (Rcpy 



9. 



(( >- 



Consequently, the equation obtains 
9' 



a (PR Qr) = : > (PR 2 + Qr 2 ) 



(1) 



Now the motion of this case is a uniformly accelerated 
motion ; consequently, the same relation obtains here 
between the angle a, the angular velocity cp, and the 



THE EXTEXSIOM OF THE PRIXCIPLES. 



345 



angular acceleration ip, as obtains in free descent be- 
tween s, v, g. If in free descent s = v 2 J'lg, then here 
a = (p 2 I'l tp. 

Introducing this value of a in equation (i), we get 
for the angular acceleration of P, ip = (PR Qrj 
PR 2 -f- Qr 2 )g, and, consequently, for its absolute ac- 
celeration y = (PR Qr/PR 2 ~+~Qr 2 ) Rg, exactly as 
in the previous treatment of the problem. ^_ 

As a second example let us consider the case of a a roiling 

... . cylinder on 

massless cylinder of radius r, in the surface of which, an inclined 

plane. 

diametrically opposite each other, are fixed two equal 
masses ///, and which in consequence of the weight of 





FiR- 174- 

these masses rolls without sliding down an inclined 
plane of the elevation a. First, we must convince our- 
selves, that in order to represent the total vis viva of 
the system we have simply to sum up the vis viva of 
the motions of rotation and progression. The axis of 
the cylinder has acquired, we will say, the velocity u 
in the direction of the length of the inclined plane, and 
we will denote by v the absolute velocity of rotation of 
the surface of the cylinder. The velocities of rotation v 
of the two masses m make with the velocity of progres- 
sion u the angles and H' (Fig. 175), where -\- 0' 
= 180. The compound velocities w and z satisfy 
therefore the equations 

w 2 = u 2 -\- v 2 2uvcosd 

z 2 = n 2 -f- v 2 'lnv cos 8' . 



346 THE SCIENCE OF MECHANICS. 

The law of But since cos 8 = cos 8', it follows that 

motion of 

finder. tt> + ** ^ 2 K + 2 tf *, Of, 

\mw 2 -f- \mz 2 = \m2u 2 -\- \m2v 2 = mu 2 -\- mv 2 . 
If the cylinder moves through the angle cp, m describes 
in consequence of the rotation the space rep, and the 
axis of the cylinder is likewise displaced a distance rep. 
As the spaces traversed are to each other, so also 
are the velocities v and //, which therefore are equal. 
The total vis viva may accordingly be expressed by 
\\- , 2mu 2 . If / is the distance the cylinder travels along 
the length of the inclined plane, the work done is 
2mg . /s'mex = 2wu 2 ; whence = Vgl- sin a. If we 
compare with this result the velocity acquired by a body 
in sliding down an inclined plane, namely, the velocity 
V igl sin a, it will be observed that the contrivance we 
are here considering moves with only one-half the ac- 
celeration of descent that (friction neglected) a sliding 
body would under the same circumstances. The rea- 
soning of this case is not altered if the mass be uni- 
formly distributed over the entire surface of the cylin- 
der. Similar considerations are applicable to the case 
of a sphere rolling down an inclined plane. It will be 
seen, therefore, that Galileo's experiment on falling 
bodies is in need of a quantitative correction. 
a modifica- Next, let us distribute the mass /;/ uniformly over 
A preceding the surface of a cylinder of radius R, which is coaxal 
with and rigidly joined to a massless cylinder of radius 
r, and let the latter roll down the inclined plane. Since 
here v/u = R /r, the principle of vis viva gives tngl 
sma = ^mu 2 (l -j- R 2 /r 2 ), whence 





THE EXTENSION OF THE PAVNC/PLES. 347 

For Rjr = 1 the acceleration of descent assumes its 
previous value g/2. For very large values of Rjr the 
acceleration of descent is very small. When R/r = 00 
it will be impossible for the machine to roll down the 
inclined plane at all. 

As a third example, we will consider the case of a The motion 
chain, whose total length is /, and which lies partly on on an in- 
a horizontal plane and partly on a plane having the plane, 
angle of elevation a. If we imagine the surface on 
which the chain _ g _ g _ gfa< _ w _ | _ grf ^___ 
rests to be very 
smooth, any very 
small portion of 
the chain left hang- 

. . Fig. 176. 

ing over on the in- 
clined plane will draw the remainder after it. If jj is 
the mass of unit of length of the chain and a portion x 
is hanging over, the principle of vis viva will give for 
the velocity v acquired the equation 

file' 2 x . x 2 . 

., = M*g g sin a = 11 g j sin a, 

or v = x ]/g sin a /I. In the present case, therefore, 
the velocity acquired is proportional to the space de- 
scribed. The very law holds that Galileo first con- 
jectured was the law of freely falling bodies. The 
same reflexions, accordingly, are admissible here as at 
page 248. 

3. Equation (1), the equation of vis viva, can always Extension 
be employed, to solve problems of moving bodies, cfpie^f"^ 
when the total distance traversed and the force that*"' 
acts in each element of the distance are known. It was 
disclosed, however, by the labors of Euler, Daniel Ber- 
noulli, and Lagrange, that cases occur in which the 



348 



THE SCIENCE OF MECHANICS. 



The re- 
searches of 
Euler. 



principle of vis viva can be employed without a knowl- 
edge of the actual path of the motion. We shall see 
later on that Clairaut also rendered important services 
in this field. 

Galileo, even, knew that the velocity of a heavy 
falling body depended solely on the vertical height de- 
scended through, and not on the length or form of the 
path traversed. Similarly, Huygens finds that the vis 
viva of a heavy material system is dependent on the 
vertical heights of the masses of 
the system. Euler was able to 
make a further step in advance. 
If a body K (Fig. 177) is at- 
tracted towards a fixed centre 
C in obedience to some given 
law, the increase of the vis viva 
in the case of rectilinear ap- 
proach is calculable from the 
initial and terminal distances 
(r Q , r,). But the increase is the 
same, if .AT passes at all from the 
position r Q to the position r independently of the 
form of its path, KB. For the elements of the work 
done must be calculated from the projections on the 
radius of the actual displacements, and are thus ulti- 
mately the same as before. 

If K is attracted towards several fixed centres C, 
baniei Ber- C , C" . . . ., the increase of its vis viva depends on the 

noulli and ...... , 

Lagrange, initial distances r Q , r , r o ". . . . and on the terminal 
distances r r', r". . . ., that is on the initial and ter- 
minal positions of K. Daniel Bernoulli extended this 
idea, and showed further that where movable bodies 
are in a state of mutual attraction the change of vis viva 
is determined solely by their initial and terminal dis- 




Fig- 177- 



The re 
searches of 



THE EXTEXSIOX OF THE PRIXCIPLES. 349 

tances from one another. The analytical treatment of 
these problems was perfected by Lagrange. If we join 
a point having the coordinates a, b, c with a point hav- 
ing the coordinates x, y, z, and denote by r the length 
of the line of junction and by a, /?, y the angles that 
line makes with the axes of x, y, z, then, according to 
Lagrange, because 

r- = (* - <0 2 + 0' - ^) 2 + (* - 2 , 

x a dr a v b dr 
cos a = = cosp = - =, 

r dx r dy 

z c dr 

COS V = s= . 

' r dz 

Accordingly, if /(r) = ~-^ is the repulsive force, or The force 

dr compo- 

the negative of the attractive force acting between the till diler- 

. -ii, ential coef- 

two points, the components will be ficiemsof 

the same 

v s, ^ dF(r)dr dF(r) function of 

X =/(r) cos a = A-= = ^. coard >- 

J v J dr dx dx ' nates - 

v /y ^ a dF{r)dr dF(r) 

dF(r) dr dF{r) 

Z =/r>) cosy = --/ _ =_U 

The force-components, therefore, are the partial 
differential coefficients of one and the same function of 
r, or of the coordinates of the repelling or attracting 
points. Similarly, if several points are in mutual ac- 
tion, the result will be 

X= ~dx 

dy 

"" dt ' 



35 THE SCIENCE OF MECHANICS. 

The force- where U is a function of the coordinates of the points. 
This function was subsequently called by Hamilton* 
the force-function. 

Transforming, by means of the conceptions here 
reached, and under the suppositions given, equation 
(i) into a form applicable to rectangular coordinates, 
we obtain 

2f(Xdx + Ydy + Zdz) = 2%m (p* v*) or, 
since the expression to the left is a complete differen- 
tial, 

where 7, is a function of the terminal values and U 
the same function of the initial values of the coordi- 
nates. This equation has received extensive applica- 
tions, but it simply expresses the knowledge that under 
the conditions designated the work done and therefore 
also the vis viva of a system is dependent on the posi- 
tions, or the coordinates, of the bodies constituting it. 
If we imagine all masses fixed and only a single 
one in motion, the work changes only as ^changes. 
The equation 7= constant defines a so-called level 
surface, or surface of equal work. Movement upon 
such a surface produces no work. U increases in the 
direction in which the forces tend to move the bodies. 



THE PRINCIPLE OF LEAST CONSTRAINT. 

i. Gauss enunciated (in Crelle's Journal fur Mather 
tnatik, Vol. IV, 1829, p. 233) a new law of mechanics, 
the principle of least constraint. He observes, that, in 

* On a General Method in Dynamics, Phil. Trans, for 1834. See also C. G. 
]. Jacobi, Vorlesungen iiber Dyn-xmik, edited by Clebsch, 1866. 



THE EXTEXSIOX OF THE PRIXCIPLES. 351 

the form which mechanics has historically assumed, dy- History of 

r j 1 c , -,., the princi- 

namics is founded upon statics, {lor example, DAlem- pie of least 

. , ... . ... ,. . _. , constraint. 

bert s principle on the principle of virtual displace- 
ments,) whereas one naturally would expect that in 
the highest stage of the science statics would appear 
as a particular case of dynamics. Now, the principle 
which Gauss supplied, and which we shall discuss in 
this section, includes both dynamical and statical cases. 
It meets, therefore, the requirements of scientific and 
logical aesthetics. We have already pointed out that this 
is also true of D'Alembert's principle in its Lagrangian 
form and the mode of expression above adopted. 
No essentially new principle, Gauss remarks, can now be 
established in mechanics ; but this does not exclude 
the discovery of new points of view, from which mechan- 
ical phenomena may be fruitfully contemplated. Such 
a new point of view is afforded by the principle of 
Gauss. 

2. Let m, m, .... be masses, connected in any man- statement 

l- 1 T>1 r * tne P 1 - 

ner with one another. I hese masse s, ufree, would , under cipie. 
the action of the forces im- 
pressed on them, describe in a 
very short element of time the 
spaces a b, a, b, . . . . ; but in 
consequence of their connec- 
tions they describe in the ?ame 
element of time the spaces a c, 

.... Now, Gauss's principle asserts, that the mo- 
tion of the connected points is such that, for the motion 
illy taken, the sum of the products of the mass of 
each material particle into the square of the distance of 
its deviation from the position it would have reached if 
free, namely m{bcy + m, (b,c,y + ....= 2m(be) 2 , is 
a minimum, that is, is smaller for the actual motion 




352 THE SCIENCE OF MECHANICS. 

than for any other conceivable motion in the same con- 
nections. If this sum, 2m(bc) 2 , is less for rest than 
for any motion, equilibrium will obtain. The principle 
includes, thus, both statical and dynamical cases. 

Definition The sum 2m(bc} 2 is called the "constraint."* In 

of "con- ; ; * ' 

mint." forming this sum it is plain that the velocities present 

in the system may be neglected, as the relative posi- 
tions of a, b, c are not altered by them. 

3. The new principle is equivalent to that of 
D'Alembert j it may be used in place of the latter ; and, 
as Gauss has shown, can also be deduced from it. The 
impressed forces carry the free mass in in an element of 
time through the space ab, the effective forces carry the 
same mass in the same time in consequence of the con- 
nections through the space ac. We resolve ab into ac 
and cb; and do the same for all the 
masses. It is thus evident that 
forces corresponding to the dis- 
'y tances c b, c,b, . . . . and propor- 
tional to mcb. m.c.b,..., do not, 
owing to the connections, become 
effective, but form with the connections an equilibrat- 
ing system. If, therefore, we erect at the terminal posi- 
tions c, c n c,, the virtual displacements cy, c, y, 

forming with cb, c, b r ... the angles 6, 6, we may 
apply, since by D'Alembert's principle forces propor- 
tional to mcb, m, e, b, are here in equilibrium, the 
principle of virtual velocities. Doing so, we shall have 

* Professor Mach's term is Abweickungssumme . The Abiveichung is the 
declination or departure from free motion, called by Gauss the Ablenkung. 
(See Duhring, Principien dcr Mechanik, 168, 169; Routh, Rigid Dynamics, 
Part I, 390-394.) The quantity 2"' (be)? is called by Gauss the Zwang; and 
German mathematicians usually follow this practice. In English, the term 
constraint is established in this sense, although it is also used with another, 
hardly quantitative meaning, for the force which restricts a body absolutely 
to moving in a certain way. Trans. 




THE EXTEXS/OX OF THE PRIXCIPLES. 353 

2 ni cb .cy COS 6^0 (l)Thededuc- 

^ tion of the 

But principle 

of least 
{by) 2 = (be) 2 + (Cy) 2 Ibc . CyCOSd, constraint. 

fb y y {be) 2 = (ey) 2 2be. eycosd, and 

2 byy2m{bey=2m(eyy22mbe.eycos0 (2) 

Accordingly, since by (1) the second member of 
the right-hand side of (2) can only be = or negative, 
that is to say, as the sum 2m(e y) 2 can never be dimin- 
ished by the subtraction, but only increased, therefore 
the left-hand side of (2) must also always be positive 
and consequently 2m(by) 2 always greater than 2 m 
. which is to say, every conceivable constraint 
from unhindered motion is greater than the constraint 
for the actual motion. 

4. The declination, be, for the very small element various 

forms in 

of time r, may, for purposes of practical treatment, be which the 

. . " . principle 

designated by s, and following Scheffler (Schlomilch's may be ex- 

_ pressed. 

Zeitschrift fur Mathematik und Physik, 1858, Vol. Ill, 
p. 197), we may remark that s = yr 2 ji, where y de- 
notes acceleration. Consequently, 2ms 2 may also be 
expressed in the forms 

r 2 t 2 t* 

2m . s . s = } 2 my . s = - 2j> .s = 2my i , 

where/ denotes the force that produces the declination 
from free motion. As the constant factor in no wise 
affects the minimum condition, we may say, the actual 
motion is always such that 

2ms* (1) 

or 

2fs (2) 

cr 

-'ILL' 2 (3) 

is a minimum. 



354 



THE SCIENCE OE MECHANICS. 



The motion 
of a wheel 



and axle, third form. 



5. We will first employ, in our illustrations, the 
Here again, as our first example, we se- 
lect the motion of a wheel and axle by 
the overweight of one of its parts 
and shall use the designations above 
frequently employed. Our problem 
is, to so determine the actual accel- 
erations y of P and y, of Q, that 

(?/g) (<? - rY + (0/r) c* - r.y 

shall be a minimum, or, since y, == 
y(r/R), so that P (g y y + 
= JV shall assume its smallest value. 




Q(g+Y-r/Ky 
Putting, to this end, 



+ r 



R 



r 

R 



0, 



we get y = (PR Qr/PR 2 + Qr 2 )Rg, exactly as in 
the previous treatments of the problem. 
Descent on As our second example, the motion of descent on 
plane? me an inclined plane may be taken. In this case we shall 

employ the first form, ~2ms 2 . 
Since we have here only to 
deal with one mass, our in- 
quiry will be directed to find- 
ing that acceleration of de- 
scent y for the plane by 
which the square of the de- 
(j 2 ) is made a minimum. By Fig. 181 we 




Fig. 181. 



clination 
have 






2 I T < 

+\r- 



* 9, 



r 2 \ . 
y-^ Ism or, 



and putting d(s 2 )/dy = 0, we obtain, omitting all 
constant factors, 2;/ 2^sin = or y = g. sina, the 
familiar result of Galileo's researches. 



THE EXTEXSIOX OF THE PRIXCIPLES. 355 

The following example will show that Gauss's prin- a case of 

ciple also embraces cases of equilibrium. On the arms num. 

a, a of a lever (Fig. 182) are hung the heavy masses 

m, m. The principle requires that m{g y} 2 -\- 

m'(g y") 2 shall be a minimum. But y' = y(a'/a). 

Further, if the masses are in- 

a q- 

versely proportional to the S I 

lengths of the lever-arms, that J[\ rh 

is to say, if mjm' = a' /a, then 
y' = y(mfm'). Conse- ' tI1 

quently, m (g y) 2 + m \g + Y **/*0 a = A r must 
be made a minimum. Putting dNjdy = 0, we get 
m (1 -f- m/m'^y = or y = 0. Accordingly, in this case 
equilibrium presents the least constraint from free mo- 
tion. 

Every neu> cause of constraint, or restriction upon New causes 

of con- 

the freedom of motion, increases the quantity of con- straim in- 

... . . . . ... crease the 

straint, but the increase is always the least possible, departure 
If two or more systems be connected, the motion of motion, 
least constraint from the motions of the unconnected 
systems is the actual motion. 

If, for example, we join together several simple 
pendulums so as to form a compound linear pendulum, 
the latter will oscillate with the motion 
of least constraint from the motion of the 
single pendulums. The simple pendulum, 
for any excursion a, receives, in the di- 
rection of its path, the acceleration g 
sma. Denoting, therefore by y sin a the 
acceleration corresponding to this excur- p . _ 
sion at the axial distance 1 on the com- 
pound pendulum, ~2.ni (g sin a r y sin a) 2 or 2m (g 
ry) 2 will be the quantity to be made a minimum. Conse- 
quently, 2m(g ry)r 0, and y = g(2 t/tr/2 mr 2 ). 




356 



THE SCIENCE OE MECHANICS. 



The problem is thus disposed of in the simplest man- 
ner. But this simple solution is possible only because 
the experiences that Huygens, the Bernoullis, and oth- 
ers long before collected, are implicitly contained in 
Gauss's principle, 
niustra- 6. The increase of the quantity of constraint, or 

tions of the J 

preceding decimation, from free motion by new causes of con- 
statement. . ' 

straint may be exhibited by the following examples. 

Over two stationary pulleys A, B, and beneath a 

movable pulley C (Fig. 184), a cord is passed, each 








D 



Fig. 185. 

extremity of which is weighted with a load P ; and on 
C a. load iP -\- p is placed. The movable pulley will 
now descend with the acceleration (J>I\P -f- P) g. But 
if we make the pulley A fast, we impose upon the 
system a new cause of constraint, and the quantity of 
constraint, or declination, from free motion will be in- 
creased. The load suspended from B, since it now 
moves with double the velocity, must be reckoned as 
possessing four times its original mass. The mova- 
ble pulley accordingly sinks with the acceleration 
(P/6P -f- P) g. A simple calculation will show that the 
constraint in the latter case is greater than in the former. 



THE EXTEXSIOX OF THE PREXCIPLES. 



357 



A number, n, of equal weights, /, lying on a smooth 
horizontal surface, are attached to n small movable 
pulleys through which a cord is drawn in the manner 
indicated in the figure and loaded at its free extremity 
with /. According as all the pulleys are movable or all 
except one prefixed, we obtain for the motive weight/, 
allowing for the relative velocities of the mas ses as r e- 
f erred top, respectively, the accelerations (4 */i + 4 n)g 
and (45) g. If all the n -\- 1 masses are movable, the 
deviation assumes the value/^/4 -+- 1, which increases 
as n, the number of the movable masses, is decreased. 






V 


w 


^^y 


* 








^^ct\ 









Fig. 186. 

7. Imagine a bodv of weight O, movable on rollers Treatment 

' * ... ofame- 

on a horizontal surface, and having an inclined plane chankai 

. problem by- 

face. On this inclined face a bodv of weight P is different 

* . mechanical 

placed. We now perceive instinctively that P will de- principles, 
scend with quicker acceleration when Q is movable 
and can give way, than it will when Q is fixed and P's 
descent more hindered. To any distance of descent // 
of P a horizontal velocity v and a vertical velocity u of 
P and a horizontal velocity iv of Q correspond. Owing 
to the conservation of the quantity of horizontal mo- 
tion, (for here only internal forces act,) we have Pv = 
Qiv, and for obvious geometrical reasons (Fig. 186) 
also 

u = (v w) tan a 

The velocities, consequently, are 



358 THE SCIENCE OF MECHANICS. 



First, by the v = -> COt a . U, 

principles _p _L_ Q ' 

of the con- ' *- 

servation of p 

momentum . _,. * f 

and of vis w p \ r\ COt a ' U ' 

viva. -* \~ \~. 

For the work Ph performed, the principle of vis 
viva gives 




Multiplying by *-, we obtain 

To find the vertical acceleration y with which the 
space h is described, be it noted that h = u 2 jiy. In- 
troducing this value in the last equation, we get 

(P + Q)sm*a 

v = - . r. 

y Psm 2 a-{- Q A 

For Q = oo, y = g sin 2 a, the same as on a sta- 
tionary inclined plane. For (? = 0, y=g, as in free 
descent. For finite values of Q = mP, we get, 

1 -f m 

since -t *> 1, 

sin 2 -f~ m 

(1 -I- w)sin 2 flr 

g> gsm 2 a. 



'. vi -j- sin 2 or 

The making of Q stationary, being a newly imposed 
cause of constraint, accordingly increases the quantity 
of constraint, or declination, from free motion. 

To obtain y, in this case, we have employed the 
principle of the conservation of momentum and the 



THE EXTENSION OF THE PRINCIPLES. 359 

principle of vis viva. Employing Gauss's principle, Second, by 
we should proceed as follows. To the velocities de- cipie of 
noted as u, v, w the accelerations y, 6, correspond. 
Remarking that in the free state the only acceleration 
is the vertical acceleration of P, the others vanishing, 
the procedure required is, to make 
P P O 

- (js r) 2 + - <5 2 + * 2 =-#" 

a minimum. As the problem possesses significance 
only when the bodies P and Q touch, that is only when 
y = (<? -{- e) tan a, therefore, also 

N=- [g(6-\- e) tan al - + P d* -f ^. 
^ Kg 

Forming the differential coefficients of this expression 
with respect to the two remaining independent vari- 
ables 6 and e, and putting each equal to zero, we ob- 
tain 

[g (d + e) tan ] Ptan a -f- PS = and 

[g (<*+ f)tanflf] ^tana+ (? = 0. 

From these two equations follows immediately 
PS () = 0, and, ultimately, the same value for y 
that we obtained before. 

We will now look at this problem from another 
point of view. The body P describes at an angle /? 
with the horizon the space s, of w T hich the horizontal 
and vertical components are v and //, while simulta- 
neously Q describes the horizontal distance w. The 
force-component that acts in the direction of s is Ps'xn ft, 
consequently the acceleration in this direction, allow- 
ing for the relative velocities of P and Q, is 
P. sin/3 

g g\s) 



360 THE SCIENCE OF MECHANICS. 

Third, by Employing the following equations which are di- 

the ex- 

tended con- rectly deducible, 

cept of mo- 

ment of in- Qw=Pp 



ertia. 



# = S COS /3 

u = v tan /?. 
the acceleration in the direction of s becomes 
Qs'm/3 
Q + Pco&p g 

and the vertical acceleration corresponding thereto is 

r " e + ^cos 2 /*' 6 ' 

an expression, which as soon as we introduce by means 
of the equation u = (v -\- iv) tan a, the angle-func- 
tions of a for those of /?, again assumes the form above 
given. By means of our extended conception of mo- 
ment of inertia we reach, accordingly, the same result 
as before. 
Fourth, by Finally we will deal with this problem in a direct 
cipies. manner. The body P does not descend on the mova- 
ble inclined plane with the vertical acceleration g, with 
which it would fall if free, but with a different vertical 
acceleration, y. It sustains, therefore, a vertical coun- 
terforce (P/g)(g y)- But as P and Q, friction 
neglected, can only act on each other by means of a 
pressure S, normal to the inclined plane, therefore 

P 

(g y) = Scosa and 



& sin a = = 
g g 



From this is obtained 



p , n Q 



THE EXTEMSIOX OF THE PRIXCIPLES. 



36i 



and by means of the equation y = (6 -f e) tan a, ulti- 
mately, as before, 

(/>+ Q)sln 2 a 



/>sin 2 + Q 



g 



Q sin a cos a 
/>sin 2 + Q* 



= 



/'sinnrcosar 



isin 2 + Q 



r 



G) 
(?) 

(3) 



If we put P= Q and ar = 4.5 , we obtain for this Discussion 
r ^ TJ of the re- 

particular case y = $<?"> * = i"> e = \& For 7*/^ = suits. 

Q/g= 1 we find the "constraint," or declination from 

free motion, to be^- a /3. If we make the inclined plane 

stationary, the constraint will be g 2 /2. If Amoved on 

a stationary inclined plane of elevation /3, where 

tan/? = y/S, that is to say, in the same path in which 

it moves on the movable inclined plane, the constraint 

would only be g 2 /S- And, in that case it would, in 

reality, be less impeded than if it attained the same 

acceleration by the displacement of Q. 

8. The examples treated will have convinced us that Gauss's 

.... , , , principle 

no substantially new insight or perception is afforded bv affords no 

. . . " newinsight 

Gauss's principle. Employing form (3) of the prin- 
ciple and resolving all the forces and accelerations in 
the mutually perpendicular coordinate-directions, giv- 
ing here the letters the same significations as in equa- 
tion (1) on page 342, we get in place of the declination, 
2 , the expression 

V 



or constraint, 2 >n y 

fx 



N=S,. 



\ + ['--v) + 



<i-')\ 



c-n 



and by virtue of the minimum condition 



dN=22i 



A '-*W+ (r. 



V\dV + 



362 THE SCIENCE OF MECHANICS. 



m 



e,\dz 



= o. 



or 2 l(X )</+( V m rf) drj+(Z m B, )dB,~\ = 0. 

Gauss's and If no connections exist, the coefficients of the (in 
bert's priii- that case arbitrary) dc;, dt), dB, severally made =0, 

ciples com- . . J . 

mutable, give the equations ot motion. But if connections do 
exist, we have the same relations between dB,, d?/, dB, 
as above in equation (i), at page 342, between 6x, dy, 
6z. The equations of motion come out the same ; as 
the treatment of the same example by D'Alembert's 
principle and by Gauss's principle fully demonstrates. 
The first principle, however, gives the equations of 
motion directly, the second only after differentiation. 
If we seek an expression that shall give by differentia- 
tion D'Alembert's equations, we are led perforce to the 
principle of Gauss. The principle, therefore, is new 
only in form and not in matter. Nor does it, further, 
possess any advantage over the Lagrangian form of 
D'Alembert's principle in respect of competency to com- 
prehend both statical and dynamical problems, as has 
been before pointed out (page 342). 
Thenhys- There is no need of seeking a mystical or metaphys- 

of theprin- /Vvz/ reason for Gauss's principle. The expression "least 
constraint" may seem to promise something of the 
sort ; but the name proves nothing. The answer to the 
question, **In what does this constraint consist ? " can- 
not be derived from metaphysics, but must be sought 
in the facts. The expression (2) of page 353, or (4) of 
page 361, which is made a minimum, represents the 
work done in an element of time by the deviation of the 
constrained motion from the free motion. This work, 
the work due to the constraint, is less for the motion 
actually performed than for any other possible motion. 



ciple. 



THE EXTEXSIOX OE THE PREXCIPLES. 



363 



Once we have recognised work as the factor deter- R6ieof the 

factor work. 

minative of motion, once we have grasped the mean- 
ing of the principle of virtual displacements to be, that 
motion can never take place except where work can be 
performed, the following converse truth also will in- 
volve no difficulty, namely, that all the work that can 
be performed in an element of time actually is per- 
formed. Consequently, the total diminution of work 
due in an element of time to the connections of the 
system's parts is restricted to the portion annulled by 
the counter-work of those parts. It is again merely a 
new aspect of a familiar fact with which we have here 
to deal. 

This relation is displayed in the very simplest cases. The foun- 

1 J dations of 

Let there be two masses m and /// at A, the one 1m- the princi- 
ple recoj- 
pressed with a force />, the other with nisabiein 

" * _ the sim- 

the force q. If we connect the two, we . _ >** piest cases, 
shall have the mass 2 /// acted on by a 
resultant force r. Supposing the spaces 
described in an element of time by the 
free masses to be represented by AC, 
A B, the space described by the con- 
joint, or double, mass will be AO = 
\ A D. The deviation, or constraint, 
is m(OB 2 -f OC 2 ). It is less than 
it would be if the mass arrived at the end of the ele- 
ment of time in M or indeed in any point lying out- 
side of B C, say N, as the simplest geometrical con- 
siderations will show. The deviation is proportional 
to the expression p 2 -\- q 2 -\- ipq cos 0/2, which in the 
case of equal and opposite forces becomes 2/ 2 , and in 
the case of equal and like-directed forces zero. 

Two forces / and q act on the same mass. The 
force q we resolve parallel and at right angles to the 




FiR. 187. 



364 THE SCIENCE OF MECHANICS. 

Even in the direction of p in r and s. The work done in an element 

principle of . 

thecompo- of time is proportional to the squares of the forces, and 

sitionof , - 

forces its if there be no connections is expressible by p 2 -4- q 2 = 

properties 

are found, p 2 -\- r 2 -|- s 2 . If now r act directly counter to the 
force/, a diminution of work will be effected and the 
sum mentioned becomes (/ r) 2 -\- s 2 . Even in the 
principle of the composition of forces, or of the mutual 
independence of forces, the properties are contained 
which Gauss's principle makes use of. This will best 
be perceived by imagining all the accelerations simul- 
taneously performed. If we discard the obscure verbal 
form in which the principle is clothed, the metaphysical 
impression which it gives also vanishes. We see the 
simple fact ; we are disillusioned, but also enlightened. 
The elucidations of Gauss's principle here presented 
are in great part derived from the paper of Schefner 
cited above. Some of his opinions which I have been 
unable to share I have modified. We cannot, for ex- 
ample, accept as new the principle which he himself 
propounds, for both in form and in import it is identical 
with the D'Alembert-Lagrangian. 

AAA XXKM1- $*tf< p.4T/ 

VIII. 
THE PRINCIPLE OF LEAST ACTION. 

Theorig- i. Maupertuis enunciated, in 1747, a principle 

scureform which he called " le principe de la moindre quant He d'ac- 

cipie of Hon," the principle of least action. He declared this 

' principle to be one which eminently accorded with the 

wisdom of the Creator. He took as the measure of 

the "action" the product of the mass, the velocity, 

and the space described, or mvs. Why, it must be 

confessed, is not clear. By mass and velocity definite 

quantities may be understood ; not so, however, by 



THE EXTENSION OF THE PRIXCIPLES. 365 

space, when the time is not stated in which the space 
is described. If, however, unit of time be meant, the 
distinction of space and velocity in the examples treated 
by Maupertuis is, to say the least, peculiar. It appears 
that Maupertuis reached this obscure expression by an 
unclear mingling of his ideas of vis viva and the prin- 
ciple of virtual velocities. Its indistinctness will be 
more saliently displayed by the details. 

2. Let us see how Maupertuis applies his principle, petermina- 

. . tion of the 

If J/, /;/ be two inelastic masses, Cand c their velocities laws of im- 
pact by this 
before impact, and u their common velocity after im- principle. 

pact, Maupertuis requires, (putting here velocities for 

spaces,) that the "action" expended in the change of 

the velocities in impact shall be a minimum. Hence, 

M{C u) 2 -{-m(c u) 2 is a minimum; that is, 

M(C u)-\- m(c w) = 0; or 

_ MC+ vie 

"~~ At + m ' 

For the impact of elastic masses, retaining the same 
designations, only substituting Fand v for the two ve- 
locities after impact, the expression M(C V) 2 -j- 
-) 2 is a minimum ; that is to say, 

J/(C V)dV+ m{c v)tlv = Q (1) 

In consideration of the fact that the velocity of ap- 
proach before impact is equal to the velocity of reces- 
sion after impact, we have 

C c = (T 7-) or 

C+ V{c+v)=0 (2) 

and 

dV ,/r = (3) 

The combination of equations (1), (2), and (3) 
readily gives the familiar expressions for V and v. 
These two cases may, as we see, be viewed as pro- 



366 THE SCIENCE OF MECHANICS. 

cesses in which the least change of vis viva by reaction 
takes place, that is, in which the least counter-work is 
done. They fall, therefore, under the principle of 
Gauss. 
Mauper- 3. Peculiar is Maupertuis's deduction of the law of 

tuis's de- 

duction of the lever. Two masses M and m (Fig. 188) rest on a 

the law of . V O y 

the lever by bar a, which the fulcrum divides into the portions 

this pnn- 

cipie. x and a x. If the bar he set in rotation, the veloci- 

ties and the spaces described will be proportional to 
the lengths of the lever-arms, and Mx 2 -f- m(a x) 2 
is the quantity to be made a minimum, that is Mx 
m (a x) = ; whence x == nia/M -\- m, a condition 

that in the case of cquilib- 

, Hum is actually fulfilled. In 

f "i criticism of this, it is to be 

^ J^ remarked, first, that masses 

Fig. jss. not subject to gravity or 

other forces, as Maupertuis 
here tacitly assumes, are always in equilibrium, and, 
secondly, that the inference from Maupertuis's deduc- 
tion is that the principle of least action is fulfilled 
only in the case of equilibrium, a conclusion which it 
was certainly not the author's intention to demonstrate. 
The correc- If it were sought to bring this treatment into ap- 
pertuiss proximate accord with the preceding, we should have 
to assume that the heavy masses M and m constantly 
produced in each other during the process the least 
possible change of vis viva. On that supposition, we 
should get, designating the arms of the lever briefly by 
a, b, the velocities acquired in unit of time by u, v, and 
the acceleration of gravity by g, as our minimum ex- 
pression, M(g u) 2 -\- m(g v) 2 ; whence M(g u) 
du -f- m(g v)dv = 0. But in view of the connection 
of the masses as lever, 



deduction. 



THE EXTEXSIOX OF THE PRINCIPLES. 



367 



// v 

-=_ 7> and 



du = 



dv\ 



whence these equations correctly follow 

Ma mb . Ma m b 



u = a 



= b 



& 



Ma* + mb**' ' ' "Ma* + mb* 
and for the case of equilibrium, where u = r = 0, 
Ma mb = 0. 

Thus, this deduction also, when we come to rectify 
it, leads to Gauss's principle. 

4. Following the precedent of Fermat and Leib- Treatment 

A r -i ii- . ' tne mo " 

nitz. Maupertuis also treats by his method the motion uon of light 

, . . , _ T , by the pnn- 

of light. Here again, however, 
he employs the notion " least ac- 
tion " in a totally different sense. 
The expression which for the 
case of refraction shall be a min- 
imum, is m . AR -f n . RB, 
where AR and RB denote the 
paths described by the light in 
the first and second media re- 
spectively, and m and n the corresponding velo- 
cities. True, we really do obtain here, if R be de- 
termined in conformity with the minimum condition, 
the result sin a /sin/? = n/m = const. But before, the 
action" consisted in the change of the expressions 
mass x velocity X distance ; now, however, it is con- 
stituted of the sum of these expressions. Before, the 
spaces described in unit of time were considered ; in 
the present case the total spaces traversed are taken. 
Should not m . AR n.RB or \m ri)(AR RB) 
be taken as a minimum, and if not, why not? But 




Fi*. 189. 



368 THE SCJEXCE OF MECHANICS. 

even if we accept Maupertuis's conception, the recip- 
rocal values of the velocities of the light are obtained, 
and not the actual values, 
character!- It will thus be seen that Maupertuis reallv had no 

sation of . 

Mauper- principle, properly speaking, but only a vague form- 
cipie. ula, which was forced to do duty as the expression of 

different familiar phenomena not really brought under 
one conception. I have found it necessary to enter- 
into some detail in this matter, since Maupertuis's per- 
formance, though it has been unfavorably criticised by 
all mathematicians, is, nevertheless, still invested with 
a sort of historical halo. It would seem almost as if 
something of the pious faith of the church had crept 
into mechanics. However, the mere endeavor to gain 
a more extensive view, although beyond the powers of 
the author, was not altogether without results. Euler, 
at least, if not also Gauss, was stimulated by the at- 
tempt of Maupertuis. 
Euler'scon- c. Euler's view is, that the purposes of the phe- 

tributions . 

to this sub- noraena of nature afford as good a basis of explana- 

ject. 

tion as their causes. If this position be taken, it will 
be presumed a priori that all natural phenomena pre- 
sent a maximum or minimum. Of what character this 
maximum or minimum is, can hardly be ascertained 
by metaphysical speculations. But in the solution of 
mechanical problems by the ordinary methods, it is 
possible, if the requisite attention be bestowed on the 
matter, to find the expression which in all cases is 
made a maximum or a minimum. Euler is thus not 
led astray by any metaphysical propensities, and pro- 
ceeds much more scientifically than Maupertuis. He 
seeks an expression whose variation put = gives the 
ordinary equations of mechanics. 

For a single body moving under the action of forces 



THE EXTEXSIOX OF THE PRIXCIPLES. 



369 



Euler finds the requisite expression in the formula The form 

which the 

ds, where ds denotes the element of the path and principle 

. ~, . . . assumed iu 

v the corresponding velocity. This expression is smaller Euier's 

hands. 

for the path actually taken than for any other infinitely 
adjacent neighboring path between the same initial 
and terminal points, which the body may be constrained 
to take. Conversely, therefore, by seeking the path that 
makes Cv ds a minimum, we can also determine the 
path. The problem of minimising Cv ds is, of course, 
as Euler assumed, a permissible one, only when v de- 
pends on the position of the elements ds, that is to 
say, when the principle of vis viva holds for the forces, 
or a force-function exists, or what is the same thing, 
when v is a simple function of coordinates. For a mo- 
tion in a plane the expression would accordingly as- 
sume the form 



>!'+() 



dx 



In the simplest cases Euier's principle is easily veri- 
fied. If no forces act, v is constant, and the curve of 
motion becomes a straight line, for which Cv ds = 
Is is unquestionably shorter than for any other 
curve between the same terminal points. 
Also, a body moving on a curved surface 
without the action of forces or friction, 
preserves its velocity, and describes on 
the surface a shortest line. 

The consideration of the motion of a 
projectile in a parabola ABC (Fig. 190) 
will also show that the quantity Cv ds 
is smaller for the parabola than for any 
other neighboring curve ; smaller, even, 
than for the straight line AB^C between the same ter- 
minal points. The velocity, here, depends solely on the 




Euier's 
principle 
applied to 
f the motion 

** of a projec- 
tile. 



Fig. 190 



T>! 



37o THE SCIENCE OF MECHANICS. 

Mathemat- vertical space described by the body, and is therefore 
opmentof the same for all curves whose altitude above OC is the 
same. If we divide the curves by a system of horizontal 
straight lines into elements which severally correspond, 
the elements to be multiplied by the same v's, though 
in the upper portions smaller for the straight line AD 
than for A B, are in the lower portions just the reverse ; 
and as it is here that the larger v's come into play, the 
sum upon the whole is smaller for^ B C than for the 
straight line. f 

Putting the origin of the coordinates at A, reckon- 
ing the abscissas x vertically downwards as positive, 
and calling the ordinates perpendicular thereto r, we 
obtain for the expression to be minimised 





where g denotes the acceleration of gravity and a the 
distance of descent corresponding to the initial velocity. 
As the condition of minimum the calculus of variations 
gives 



'i*&+*>& 



= Cot 



v+(% 



dy C 

-y = or 

ax \/2g{a + x)C* 

Cdx 



Ji/2 



1/2J- (* + *) C* 

and, ultimately, 

C 

y =-Vlg(a + x) c 2 -f c > 



THE EXTENSION OF THE PRINCIPLES. 371 

where C and C denote constants of integration that 

pass into C= 1 2ga and C'= 0, if for x = 0, dx/dy = 

and y = be taken. Therefore, y = 2 V / fjr. By this vf 

method, accordingly, the path of a projectile is shown 

to be of parabolic form. 

6. Subsequently, Lagrange drew express attention The addi- 

1 r 1 t- 1 1 t 1 1 1 tions of La- 

to the fact that Euler s principle is applicable only in grange and 
cases in which the principle of vis viva holds. Jacobi 
pointed out that we cannot assert that Cv ds for the ac- 
tual motion is a minimum, but simply that the variation of 
this expression, in its passage to an infinitely adjacent 
neighboring path, is = 0. Generally, indeed, this con- 
dition coincides with a maximum or minimum, but it 
is possible that it should occur without such ; and the 
minimum property in particular is subject to certain 
limitations. For example, if a body, constrained to 
move on a spherical surface, is set in motion by some 
impulse, it will describe a great circle, generally a 
shortest line. But if the length of the arc described 
exceeds 180 , it is easily demonstrated that there exist 
shorter infinitely adjacent neighboring paths between 
the terminal points. 

7. So far, then, this fact only has been pointed out, Eniers 
that the ordinary equations of motion are obtained bv but one of 

__.-.i , / , _, . ' manv which 

equating the variation of Cv ds to zero. But since the give the 

fi *-r i i- r 1 1 equations 

properties of the motion of bodies or of their paths may of motion, 
always be defined by differential expressions equated 
to zero, and since furthermore the condition that the 
variation of an integral expression shall be equal to 
zero is likewise given by differential expressions equated 
to zero, unquestionably various other integral expres- 
sions may be devised that give by variation the ordi- 
nary equations of motion, without its following that the 



372 THE SCIENCE OF MECHANICS. 

integral expressions in question must possess on that 
account any particular physical significance. 
Yet the ex- 8. The striking fact remains, however, that so simple 

pression . 

must pos- an expression as fv as does possess the property men- 
icai import, tioned, and we will now endeavor to ascertain its phys- 
ical import. To this end the analogies that exist be- 
tween the motion of masses and the motion of light, as 
well as between the motion of masses and the equilib- 
rium of strings analogies noted by John Bernoulli 
and by Mobius will stand us in stead. 

A body on which no forces act, and which there- 
fore preserves its velocity and direction constant, de- 
scribes a straight line. A ray of light passing through 
a homogeneous medium (one having everywhere the 
same index of refraction) describes a straight line. A 
string, acted on by forces at its extremities only, as- 
sumes the shape of a straight line. 
Elucidation A body that moves in a curved path from a point 

i t this im- . , . . 

ponbytiie A to a. point B and whose velocity v = cp(x, y, z) is a 
mass, the function of coordinates, describes between A and B a 

motion of a . , ^ . . . . 

ray of light, curve for which generally f p as is a minimum. A ray 
equilibrium of light passing from A to B describes the same curve, 
if the refractive index of its medium, = <p(jx, y 
is the same function of coordinates ; and in this case 
Cuds is a minimum. Finally, a string passing from 
A to B will assume this curve, if its tension S = 
cp (x, y, z) is the same above-mentioned function of co- 
ordinates ; and for this case, also, CSds is a minimum. 
The motion of a mass may be readily deduced from 
the equilibrium of a string, as follows. On an element 
ds of a string, at its two extremities, the tensions S, S' 
act, and supposing the force on unit of length to be P, 
in addition a force P. ds. These three forces, which 
we shall represent in magnitude and direction by BA, \ 



THE EXTEXSIOX OF THE PRIXCIPLES. 



373 



BC, BD (Fig. 191), are in equilibrium. If now, a body, The motion 
with a velocity v represented in magnitude and direc- deduced 
lion by AB, enter the element of the path ds, and re- equilibrium 

- 1 1 1 r> 7- of a string. 

ceive within the same the velocity component BP 
BD, the body will proceed on- 
ward with the velocity v' = BC. 
Let Q be an accelerating force 
whose action is directly opposite 
to that of P; then for unit of time 
the acceleration of this force will 
be Q, for unit of length of the 
string Q/r, and for the element 
of the string (Q/v)ds. The body will move, therefore, 
in the cur: c of the string, if we establish between the 
forces P and the tensions S, in the case of the string, 
and the accelerating forces Q and the velocity v in the 
case of the mass, the relation 




Fig. 191. 



S-.v. 



The minus sign indicates that the directions of P and 
Q are opposite. 

A closed circular string is in equilibrium when be- The equi- 

librrtim of 

tween the tension S of the string, everywhere constant, closed 

. strings. 

and the force P falling radially outwards on unit of 
length, the relation P = S/r obtains, where r is the 
radius of the circle. A body will move with the con- 
stant velocity r in a circle, when between the velocity 
and the accelerating force Q acting radially inwards 
the relation 



= - or Q = obtains. 

r r 



A body will move with constant velocity v in any curve 
when an accelerating force Q r- r constantly acts 



374 



THE SCIENCE OE MECHANICS. 



on it in the direction of the centre of curvature of each 
element. A string will lie under a constant tension S 
in any curve if a force P = S/r acting outwardly from 
the centre of curvature of the element is impressed on 
unit of length of the string. 
The deduc- No concept analogous to that of force is applicable 

tion of the . r 

motion of to the motion of light. Consequently, the deduction of 

light from . * J 

tho motions the motion of light from the equilibrium of a string or 

of masses . 

and the the motion of a mass must be differently effected. A 

equilihrium _ * 

of strings, mass, let us say, is moving with the velocity AB -. v. 

(Fig. 192.) A force in the direction 
BD is impressed on the mass which 
produces an increase of velocity BE, 

A \B $ so that by the composition of the ve- 
locities BC = AB and BE the new 
velocity BE= v is produced. If we 
resolve the velocities v, v' into com- 
ponents parallel and perpendicular to 
the force in question, we shall per- 
ceive that the parallel components alone 
are changed by the action of the force. 
This being the case, we get, denoting 

by k the perpendicular component, and by a and a' 

the angles v and v' make with the direction of the 

force, 

k = v sin a 

k = v' sin a or 





V 


K 


G 




c 


E 
D 




P 



Fig. 192. 



sin a- 
sin a-' 



If, now, we picture to ourselves a ray of light that 
penetrates in the direction of v a refracting plane at 
right angles to the direction of action of the force, and 
thus passes from a medium having the index of refrac- 



THE EXTEXSWX OF THE FR/XCIPLES. 375 

tion n into a medium having the index of refraction d, Develop- 

, , , , , . , i- 1 -ii . 1 meat of this 

where n/n =7- r, this ray of light will describe the illustration, 
sane path as the body in the case above. If, there- 
fore, we wish to imitate the motion of a mass by the 
motion of a ray of light (in the same curve), we must 
everywhere put the indices of refraction, n, proportional 
to the velocities. To deduce the indices of refraction 
from the forces, we obtain for the velocity 



a \ \ = Pdq, and 
for the index of refraction, by analogy, 



where P denotes the force and dq a distance-element 
in the direction of the force. If ds is the element of 
the path and a the angle made by it with the direction 
of the force, we have then 



(8- 



Pcosa . ds 



a = / cos a .ds. 

For the path of a projectile, under the conditions above 
assumed, we obtained the expression y = 2 1 ax. This 
same parabolic path will be described by a ray of light, 
if the law n = l''2g(a x) be taken as the index of 
refraction of the medium in which it travels. 

9. We will now more accurately investigate the Relation of 
manner in which this minimum property is related to mum prop- 
the form of the curve. Let us take, first, (Fig. 193) aforin'of e 
broken straight line ABC, which intersects the straight curve *' 
line MN t put AB = s; BC=s', and seek the condition 
that makes vs v's' a minimum for the line that passes 



376 



THE SCIENCE OE MECHANICS. 



First, de- through the fixed points A and B, where v and v are 

duction of 

the mini- supposed to have different, though constant, values 

mum condi- 

tion. above and below MN. If we displace the point B an 

infinitely small distance to D, the new line through A 
and Cwill remain parallel to the original one, as the 
drawing symbolically shows. The expression v s -|- r's' 
is increased hereby by an amount 

v m sin a -j- v m sin a, 
where m = DB. The alteration is accordingly propor- 
tional to psinflf + p'sina', and the condition of 
minimum is that 



p'sinflr' = 1 or 



sin a 
sin a' 




Fig. 193. 



B N 




Fig. 19;. 



If the expression s/<' -(- s jv' is to be made a minimum, 
we have, in a similar way, 



sin a 
sin or' 



Second, the If, next, we consider the case of a string stretched 

application . , .. . . ,-. _ . . . i-ir. jo' 

of this con- in the direction AB C, the tensions 01 which o and o 
equilibrium are different above and below MN, in this case it is 
the minimum oi Ss -\- S's' that is to be dealt with. To 
obtain a distinct idea of this case, we may imagine the 



THE EXTEXSIOX OF THE PRIXCIPLES. 



377 




motion of a 
ray of light. 



string stretched once between A and B and thrice be- 
tween B and C, and finally a weight P attached. Then 
S = P and S' = 3 P. If we displace the point B a dis- 
tance ///, any diminution of the expression Ss -\- S's' 
thus effected, will express the increase of work which 
the attached weight P performs. If 5/// sin + 
S'm sin a' = 0, no work is performed. Hence, the mini- 
mum of Ss -\- S's' corresponds to a maximum of work. 
In the present case the principle of least action is sim- 
ply a different form of the principle of virtual displace- 
ments. 

Now suppose that ABC is a ray of light, whose ve- Third, the 

" application 

locities r and v above and below MN are to each other of this con- 

_. . /!,. ditiontothe 

as 3 to 1. The motion of light be- 
tween two points A and B is such 
that the light reaches B in a mini- 
mum of time. The physical reason 
of this is simple. The light travels 
from A to B, in the form of ele- 
mentary waves, by different routes. 
Owing to the periodicity of the light, 
the waves generally destroy each 
other, and only those that reach the 
designated point in equal times, that is, in equal phases, 
produce a result. But this is true only of the waves 
that arrive by the minimum path and its adjacent neigh- 
boring paths. Hence, for the path actually taken by 
the light s v / v is a minimum. And since the in- 
dices of refraction n are inversely proportional to the 
velocities v of the light, therefore also ns -f- n's' is a 
minimum. 

In the consideration of the motion of a mass the con- 
dition that vs v's' shall be a minimum, strikes us as 
something novel. (Fig. 195.) If a mass, in its passage 



Fig- 195- 



378 THE SCIENCE OF MECHANICS. 

Fourth, it* through a plane MN, receive, as the result of the action 

application . . ,.,,. . r^ r, 

to the mo- of a force impressed in the direction DB, an increase of 
mass. velocity, by which v, its original velocity, is made v' , we 

have for the path actually taken by the mass the equa- 
tion V sin (X = 7'' sin a' = k. T/iis equation, which is also 
the condition of minimum, simply states that only the ve- 
locity-component parallel to the direction of the force is 
altered, but that the component k at right angles thereto re- 
mains unchanged. Thus, here also, Euler's principle 
simply states a familiar fact in a new form. (See p. 575.) 
Formofthe io. The minimum condition v sin a -\- if sin *= 

minimum . . f . r . . , 

condition may also be written, if we pass from a finite broken 

applicable .... . . . . . r 

to curves, straight line to the elements of curves, in the form 
v sin a -f- (?' + dv) sin(ar -f- da) = 
or 

d(v sin a) = 

or, finally, 

v sin a = const. 

In agreement with this, we obtain for the motion 

of light 

d (n sin a) = 0, n sin a = const, 

sin** 

= 0, = const, 

v 

and for the equilibrium of a string 

d (S sin a) = 0, *S sin a = const. 

To illustrate the preceding remarks by an ex- 
ample, let us take the parabolic path of a projectile, 
where a always denotes the angle that the element of 
the path makes with the perpendicular. Let the ve- 
locity be v = V 2g{a -\- x), and let the axis of the y-ox- 
dinates be horizontal. The condition v . sin a = const, 
or V-zg{a-\- x). dy/ds = const, is identical with that 
which the calculus of variation gives, and we now know 



THE EX TEX SI OX OF THE PRIXCIPLES. 



379 




Fir. 196. 



its simple physical significance. If we picture to ourselves illustration 

, . / of the three 

a string whose tension is S = V 2f(a -f- x), an arrange- typical 

cast - 

ment which might be effected bv fixing frictionless curvilinear 

motions. 

pulleys on horizontal parallel rods placed in a vertical 
plane, then passing the string through these a sufficient 
number of times, and finally attaching 
a weight to the extremity of the string, 
we shall obtain again, for equilibrium, 
the preceding condition, the phys- 
ical significance of which is now ob- 
vious. When the distances between 
the rods are made infinitely small the 
string assumes the parabolic form. 
In a medium, the refractive index of 
which varies in the vertical direction 
by the law n = V 2g{a -f- x), or the velocity of light in 
which similarly varies by the law v = \/V 2g{a -f- x), 
a ray of light will describe a path which is a parabola. 
If we should make the velocity in such a medium 
7' = y2g(a-{-x'), the ray would describe a cycloidal path, 
for which, not C\' 2g(a -f- x) . ds, but the expression 
\ / 2g{a -j- x) would be a minimum. 
11. In comparing the equilibrium of a string with 
the motion of a mass, we may employ in place of a 
string wound round pulleys, 
a simple homogeneous cord, 
provided we subject the cord 
to an appropriate system of 
forces. We readily observe 
that the systems of forces 
that make the tension, or, 
as the case may be, the ve- 
locity, the same function of coordinates, are differ- 
ent. If we consider, for example, the force of gravity, 




Fig. 197. 



3^o THE SCIENCE OF MECHANICS. 



"he condi- v = \/ ig(a 4- x). A string, however, subjected to the 

tions and ... 

conse- action of gravity, forms a catenary, the tension of 

quences of . . 

thepreced- which is given by the formula S = m nx, where m 

ing analo- 
gies, and // are constants. The analogy subsisting between 

the equilibrium of a string and the motion of a mass is 
substantially conditioned by the fact that for a string 
subjected to the action of forces possessing a force- 
function U, there obtains in the case of equilibrium 
the easily demonstrable equation U -\- S = const. This 
physical interpretation of the principle of least action 
is here illustrated only for simple cases ; but it may 
also be applied to cases of greater complexity, by 
imagining groups of surfaces of equal tension, of equal 
velocity, or equally refractive indices constructed which 
divide the string, the path of the motion, or the path 
of the light into elements, and by making a in such a 
case represent the angle which these elements make 
with the respective surface-normals. The principle of 
least action was extended to systems of masses by La- 
grange, who presented it in the form 

6 ^ 7ii Cv ds 0. 
If we reflect that the principle of vis viva, which is the 
real foundation of the principle of least action, is not 
annulled by the connection of the masses, we shall 
comprehend that the latter principle is in this case also 
valid and physically intelligible. 



HAMILTON'S PRINCIPLE. 

i. It was above remarked that various expressions 
can be devised whose variations equated to zero give 
the ordinary equations of motion. An expression of 
this kind is contained in Hamilton's principle 



THE EXTENSION OF THE PRINCIPLES. 



38i 



8J(U+ T)di = 0, or 

'o 

f{6U+ ST)dt = 0, 



to 

where St/and 3 T denote the variations of the work 
and the vis viva, vanishing for the initial and terminal 
epochs. Hamilton's principle is easily deduced from 
mbert's, and, conversely, D'Alembert's from 
Hamilton's ; the two are in fact identical, their differ- 
ence being merely that of form.* 

2. We shall not enter here into any extended in- 
vestigation of this subject, but simply exhibit the iden- 
tity of the two principles by an example 
the same that served to illustrate the prin- 
ciple of D'Alembert: the motion of awheel 
and axle by the over-weight of one of its 
parts. In place of the actual motion, we 
may imagine, performed in the same inter- 
val of time, a different motion, varying in- 
finitely little from the actual motion, but 
coinciding exactly with it at the beginning 
and end. There are thus produced in every element 
of time dt, variations of the work (6 V) and of the vis 
viva (6T); variations, that is, of the values {/and T 
realised in the actual motion. But for the actual mo- 
tion, the integral expression, above stated, is = 0, and 
may be employed, therefore, to determine the actual 
motion. If the angle of rotation performed varies in 
the element of time dt an amount a from the angle of 
the actual motion, the variation of the work corre- 
sponding to such an alteration will be 

6 U {PR Q r )a = Ma. 

* Compare, for example, Kirchhoff, Vm-lesungen &ber mathrmatische Phy- 
stk, Meckanik, p. 25 et stgq., and Jacobi, VorUsungen ttber Dynamik, p. 58. 



The points 
of identity 
of Hamil- 
ton's and 
D'Alem- 
bert's prin- 
ciples. 




Hamilton's 
principle 
applied to 
the motion 
of a wheel 
and axle. 



Fig. 198. 



382 THE SCIENCE OE MECHANICS. 

Mathemat- The vis viva, for any given angular velocity oo, is 

ical devel- 
opment of .**-. ~ CO 2 
this case. T= - {PR 2 + Qr*) = . 

and for a variation do? of this velocity the variation of 
the vis viva is 

<*> 

But if the angle of rotation varies in the element dt an 
amount a, 

6co= . and 

d t 



The form of the integral expression, accordingly, is 

j/ _|_ N d<X 
But as 



* i 

/ 



dt 

to - 



dt = 0. 



<// v ; dt ^ dt' 



therefore, 



ti 



The second term of the left-hand member, though, 
drops out, because, by hypothesis, at the beginning 
and end of the motion a = 0. Accordingly, we have 

"'m- ) >it = 0, 

an expression which, since a in every element of time 
is arbitrary, cannot subsist unless generally 

m- <= o. 

dt 



- 1 

f 



THE EXTEXSIOX OF THE PRIXCIPLES. 383 

Substituting for the symbols the values they represent, 
we obtain the familiar equation 

doo _ PR Qr 
~dl ~~ PR- Qr* 8 ' 
D'Alembert's principle gives the equation The same 

results ob- 
f V*\ tained by 

M \a = 0, <he use of 

dt I * DAlem- 

/ bert's prin- 

which holds for every possible displacement. We might, ciple " 
in the converse order, have started from this equation, 
have thence passed to the expression 

'o 

and, finally, from the latter proceeded to the same re- 
sult 



/ 



Ma + N d -\ dt (Na) X = 
'1 

3. As a second and more simple example let us illustration 
consider the motion of vertical descent. For every by thl^o-' 
infinitely small displacement s the equation subsists ocai de- 
[**g m(dv/dty\s = 0, in which the letters retain s 
their conventional significance. Consequently, this 
equation obtains 

J U'gm^)s.dt=0, 

which, as the result of the relations 
,(r/i7's) dv , ds 

4 dt' = m ji S + mV Ji* 



384 THE SCIENCE OF MECHANICS. 

U 

.. ' dt = (m v s) = 0, 

ltt to 

l o 

provided s vanishes at both limits, passes into the form 
'i 

that is, into the form of Hamilton's principle. 

Thus, through all the apparent differences of the 
mechanical principles a common fundamental same- 
ness is seen. These principles are not the expression 
of different facts, but, in a measure, are simply views 
of different aspects of the same fact. 



SOME APPLICATIONS OF THE PRINCIPLES OF MECHANICS TO 
HYDROSTATIC AND HYDRODYNAMIC QUESTIONS. 

Method of i. We will now supplement the examples which 

fh e m aTtion g we have given of the application of the principles 
on liquid 7 of mechanics, as they applied to rigid bodies, by a 
few hydrostatic and hydrodynamic illustrations. We 
shall first discuss the laws of equilibrium of a weightless 
liquid subjected exclusively to the action of so-called 
molecular forces. The forces of gravity we neglect in 
our considerations. A liquid may, in fact, be placed 
in circumstances in which it will behave as if no forces 
of gravity acted. The method of this is due to Pla- 
teau.* It is effected by immersing olive oil in a mix- 
ture of water and alcohol of the same density as the 
oil. By the principle of Archimedes the gravity of the 
masses of oil in such a mixture is exactly counterbal- 
anced, and the liquid really acts as if it were devoid of 
weight. 

* Statique experimentale et thkorique des liquides, 1873. 



masses. 



THE EXTEXSIOX OF THE PRIXCIPLES. 385 

2. First, let us imagine a weightless liquid mass The work of 
free in space. Its molecular forces, we know, act only forces de- 

.... <r-i- t- it- pendent on 

at very small distances. Taking as our radius the dis- a change in 
tance at which the molecular forces cease to exert a superficial 
measurable influence, let us describe about a particle 
a, b, c in the interior of the mass a sphere the so- 
called sphere of action. This sphere of action is regu- 
and uniformly filled with other particles. The 
resultant force on the central particles a, b, c is there- 
fore zero. Those parts only that lie at a distance from 
the bounding surface less than the radius of the sphere 
of action are in different dynamic conditions from the 
particles in the interior. If the radii of curvature of 





Fig. 199- Fig. 300. 

the surface-elements of the liquid mass be all regarded 
as very great compared with the radius of the sphere 
of action, we may cut off from the mass a superficial 
stratum of the thickness of the radius of the sphere of 
action in which the particles are in different physical 
conditions from those in the interior. If we convey 
a particle a in the interior of the liquid from the posi- 
tion a to the position b or c, the physical condition 
of this particle, as well as that of the particles which 
take its place, will remain unchanged. No work can 
be done in this way. Work can be done only when a 
particle is conveyed from the superficial stratum into 
the interior, or, from the interior into the superficial 
stratum. That is to say, work can be done only by a 



/ o 



386 THE SCIENCE OF MECHANICS. 

change of size of the surface. The consideration whether 
the density of the superficial stratum is the same as 
that of the interior, or whether it is constant through- 
out the entire thickness of the stratum, is not primarily- 
essential. As will readily be seen, the variation of the 
surface-area is equally the condition of the perform- 
ance of work when the liquid mass is immersed in a 
second liquid, as in Plateau's experiments. 
Diminution We now inquire whether the work which by the 

of super- . ... ..... 

nciai area transportation of particles into the interior effects a 

due to posi- .... 

the work, diminution of the surface-area is positive or negative, 
that is, whether work is performed or work is ex- 
pended. If we put two fluid drops in contact, they 
will coalesce of their own accord ; 
and as by this action the area 
of the surface is diminished, it 
follows that the work that pro- 
duces a diminution of superfi- 
Fig 20I ' cial area in a liquid mass is posi- 

tive. Van der Mensbrugghe has 
demonstrated this by a very pretty experiment. A 
square wire frame is dipped into a solution of soap and 
water, and on the soap-film formed a loop of moistened 
thread is placed. If the film within the loop be punc- 
tured, the film outside the loop will contract till the 
thread bounds a circle in the middle of the liquid sur- J 
face. But the circle, of all plane figures of the same 
circumference, has the greatest area ; consequently, ' 
the liquid film has contracted to a minimum. 
Consequent The following will now be clear. A weightless j 
of liquid liquid, the forces acting on which are molecular forces, 

equilibrium . . . . . . 

will be in equilibrium in all forms in which a system 01 
virtual displacements produces no alteration of the 
liquid's superficial area. But all infinitely small changes 



THE EXTENSION OF THE PRINCIPLES 387 

of form may be regarded as virtual which the liquid 
admits without alteration of its volume. Consequently, 
equilibrium subsists for all liquid forms for which an 
infinitely small deformation produces a superficial va- 
riation = 0. For a given volume a minimum of super- 
ficial area gives stable equilibrium ; a maximum un- 
stable equilibrium. 

Among all solids of the same volume, the sphere 
has the least superficial area. Hence, the form which 
a free liquid mass will assume, the form of stable equi- 
librium, is the sphere. For this form a maximum of 
work is done ; for it, no more can be done If the 
liquid adheres to rigid bodies, the form assumed is de- 
pendent on various collateral conditions, which render 
the problem more complicated. 

3. The connection between the size and the form of Modeofde- 

jfii ... termining 

the liquid surface may be investigated as follows. We the connec- 

. J tion of the 

imagine the closed outer sur- 
face of the liquid to receive 
without alteration of the li- 
quid's volume an infinitely 
small variation. By two sets of 
mutually perpendicular lines 

r Fig. 202. 

ot curvature, we cut up the 

original surface into infinitely small rectangular ele- 
ments. At the angles of these elements, on the original 
surface, we erect normals to the surface, and determine 
thus the angles of the corresponding elements of the 
varied surface. To every element tlO of the original 
surface there now corresponds an element dO' of the 
varied surface ; by an infinitely small displacement, 6 n, 
along the normal, outwards or inwards, dO passes into 
dO' and into a corresponding variation of magnitude. 
Let dp, dq be the sides of the element dO. For the 




388 



THE SCIENCE OE MECHANICS. 



The mathe- sides dp', dq of the element dO', then, these relations 

matical de- 
velopment obtain 
of this 

method. / fi ' 

dp' = dp f 1 + 



where r and r' are the radii of curvature of the princi- 
pal sections touching the elements of the lines of cur- 
vature/, q, or the so-called principal radii of curva- 
ture.* The radius of curvature of an outwardly convex 
element is reckoned as positive, that of an outwardly 
concave element as negative, in the usual manner. For 
the variation of the element we obtain, accordingly, 



6.dO = dO' dO 



, $ "\ ( -, , $ n 



dpdq. 




r J \ r 

Neglecting the higher powers of Sn we 
get 

d .d0 = (- + -,]. dO. 

The variation of the whole surface, 
then, is expressed by 
1 



60 



Pi 



,+, 



dn.dO 



1) 



Fig. 203. 



Furthermore, the normal displacements 
must be so chosen that 

Cd;i.dO = (2) 

that is, they must be such that the sum of the spaces 
produced by the outward and inward displacements of 

* The normal at any point of a surface is cut by normals at infinitely neigh- 
boring points that lie in two directions on the surface from the original point, 
these two directions being at right angles to each other ; and the distances 
from the surface at which these normals cut are the two principal, or extreme, 
radii of curvature of the surface. Trans. 



THE EXTENSION OF THE PRINCIPLES. 389 

the superficial elements (in the latter case reckoned as 
negative) shall be equal to zero, or the volume remain 
constant. 

Accordingly, expressions (1) and (2) can be put a condition 
simultaneously = only if i/r + i/r' has the same va lue the generai- 

-' . j- ltyoftheex- 

for all points of the surface. This will be readily seen pressions 

r obtained, 

from the following consideration. Let the elements depends. 
dO of the original surface be symbolically represented 
by the elements of the line AX (Fig. 204) and let the 
normal displacements 6 n be erected as ordinates 
thereon in the plane E, the outward displacements up- 
wards as positive and the inward displacements down- 
wards as negative. 
Join the extremities 

of these ordinates so 
as to form a curve, 
and take the quadra- 
ture of the curve, 
reckoning the sur- 
face above AX as positive and that below it as nega- 
tive. For all systems of Sn for which this quadra- 
ture = 0, the expression (2) also = 0, and all such 
systems of displacements are admissible, that is, are 
virtual displacements. 

Now let us erect as ordinates, in the plane ', the 
values of 1 /- -f 1 /-' that belong to the elements dO. A 
case may be easily imagined in which the expressions 
(1) and (2) assume coincidently the value zero. Should, 
however, i/r-f i/V have different values for different 
elements, it will always be possible without altering 
the zero-value of the expression (2), so to distribute 
the displacements 6 n that the expression (1) shall be 
different from zero. Only on the condition that 1 /r + 
1 r has the same value for all the elements, is expres- 




Fi*. 204. 



39Q THE SCIENCE OF MECHANICS. 

sion (i) necessarily and universally equated to zero 
with expression (2). 
The sum Accordingly, from the two conditions (1) and (2^ if 

which for \ / \ j 

equilibrium follows that 1 lr + \/r'= const : that is to say, the sum 

must be ' * J 

constantforof the reciprocal values of the principal radii of curva- 

the whole . 

surface. ture, or of the radii of curvature of the principal nor- 
mal sections, is, in the case of equilibrium, constant 
for the whole surface. By this theorem the dependence 
of the area of a liquid surface on its superficial form is 
defined. The train of reasoning here pursued was 
first developed by Gauss,* in a much fuller and more 
special form. It is not difficult, however, to present 
its essential points in the foregoing simple manner. 

Application 4. A liquid mass, left wholly to itself, assumes, as 

of this gen- ' . " 

erai condi- we have seen, the spherical form, and presents an ab- 

tion to . . . 

interrupted solute minimum of superficial area. The equation 

liquid mas- . 

ses. \ jr -(- \jr' === const is here visibly fulfilled in the form 

2/R = const, R being the radius of the sphere. If the 
free surface of the liquid mass be bounded by two solid 
circular rings, the planes of which are parallel to each 
other and perpendicular to the line joining their mid- 
dle points, the surface of the liquid mass will assume 
the form of a surface of revolution. The nature of the 
meridian curve and the volume of the enclosed mass 
are determined by the radius of the rings R, by the 
distance between the circular planes, and by the value 
of the expression 1/r -f- 1/r' for the surface of revolu- 
tion. When 

I+I-I+i-I 

r r r oo R 

the surface of revolution becomes a cylindrical surface. 
For 1/r -4- l/r'= 0, where one normal section is con- 

* Principia Generalia Theories Figure? Fluidorum in Statu sEquilibrii 
Gottingen, 1830 ; Werke, Vol. V, 29, GOttingen, 1867. 



THE EXTEXSIOX OF THE PRINCIPLES. 391 

vex and the other concave, the meridian curve assumes 
the form of the catenary. Plateau visibly demonstrated 
these cases by pouring oil on two circular rings of wire 
fixed in the mixture of alcohol and water above men- 
tioned. 

. let us picture to ourselves a liquid mass Liquid mas- 
ses whose 
bounded by surface-parts for which the expression surfacesare 

. . partly con- 

1 r -}- 1 lr has a positive value, and by other parts cave and 

1 - 1 , , partly con- 

for which the same expression has a negative value, vex 
or, more briefly expressed, by convex and concave sur- 
faces. It will be readily seen that any displacement 
of the superficial elements outwards along the normal 
will produce in the concave parts a diminution of the 
superficial area and in the convex parts an increase. 
Consequently, work is performed when concave surfaces 
move outwards and convex surfaces inwards. Work 
also is performed when a superficial portion moves 
outwards for which \jr -\- 1/r' = -f a, while simulta- 
neously an equal superficial portion for which 1 r -J- 
1 >' ~> a moves inwards. 

Hence, when differently curved surfaces bound a 
liquid mass, the convex parts are forced inwards and 
the concave outwards till the condition 1/r-f- V r ' 
const is fulfilled for the entire surface. Similarly, when 
a connected liquid mass has several isolated surface- 
parts, bounded by rigid bodies, the value of the ex- 
pression 1/r -f- 1/r' must, for the state of equilibrium 
be the same for all free portions of the surface. 

For example, if the space between the two circular Experi- 

1 . , , mental 

rings in the mixture of alcohol and water above re- illustration 
ferred to, be filled with oil, it is possible, by the use conditions, 
of a sufficient quantity of oil, to obtain a cylindrical 
surface whose two bases are spherical segments. The 
curvatures of the lateral and basal surfaces will accord- 



392 THE SCIENCE OF MECHANICS. 

ingly fulfil the condition \jR -f- 1/co = 1/p + l/p, or 
p = 2R, where p is the radius of the sphere and J? that 
of the circular rings. Plateau verified this conclusion 
by experiment. 
Liquidmas- 5. Let us now study a weightless liquid mass which 

ses cnclos- 

in* a hoi- encloses a hollow space. The condition that 1/r -j- 1/r 
shall have the same value for the interior and exterior 
surfaces, is here not realisable. On the contrary, as 
this sum has always a greater positive value for the 
closed exterior surface than for the closed interior sur- 
face, the liquid will perform work, and, flowing from 
the outer to the inner surface, cause the hollow space 
to disappear. If, however, the hollow space be occu- 
pied by a fluid or gaseous substance subjected to a de- 
terminate pressure, the work done in the last-men- 
tioned process can be counteracted by the work ex- 
pended to produce the compression, and thus equilib- 
rium may be produced. 
The me- Let us picture to ourselves a liquid mass confined 

properties between two similar and similarly situated surfaces 
very near each other. A bubble is such 
a system. Its primary condition of equi- 
librium is the exertion of an excess of 
pressure by the inclosed gaseous con- 
tents. If the sum 1/r -f- 1/r' has the 
value -j- a for the exterior surface, it will 
have for the interior surface very nearly 
a. A bubble, left wholly to itself, will al- 
ways assume the spherical form. If we conceive such 
a spherical bubble, the thickness of which we neglect, 
the total diminution of its superficial area, on the 
shortening of the radius r by dr, will be 16 m dr. If, 
therefore, in the diminution of the surface by unit 
of area the work A is performed, then A . i6mdr will 



of bubbles. 




THE EXTENSION OF THE PRLXCIPLES. 393 

be the total amount of work to be compensated for 
by the work of compression p.\r 2 7tdr expended by 
the pressure / on the inclosed contents. From this 
follows \A/r = p ; from which A may be easily calcu- 
lated if the measure of r is obtained and p is found by 
means of a manometer introduced in the bubble. 

An open spherical bubble cannot subsist. If an Open 
open bubble is to become a figure of equilibrium, the 
sum 1/r+l / ' must not only be constant for each of 
the two bounding surfaces, but must also be equal for 
both. Owing to the opposite curvatures of the sur- 
faces, then, 1/r-f l/r' = 0. Consequently, r = r' 
for all points. Such a surface is called a minimal sur- 
face ; that is, it has the smallest area consistent with 
its containing certain closed contours. It is also a sur- 
face of zero-sum of principal curvatures ; and its ele- 
ments, as we readily see, are saddle-shaped. Surfaces 
of this kind are obtained by constructing closed space- 
curves of wire and dipping the wire into a solution of 
soap and water.* The soap-film assumes of its own 
accord the form of the curve mentioned. 

6. Liquid figures of equilibrium, made up of thin Plateau's 
films, possess a peculiar property. The work of the ur^'ofequi- 
forces of gravity affects the entire mass of a liquid ; ' 
that of the molecular forces is restricted to its super- 
ficial film. Generally, the work of the forces of grav- 
ity preponderates. But in thin films the molecular 
forces come into very favorable conditions, and it is 
possible to produce the figures in question without 
difficulty in the open air. Plateau obtained them by 
dipping wire polyhedrons into solutions of soap and 
water. Plane liquid films are thus formed, which meet 

The mathematical problem of determining such a surface, when the 
forms of the wires are given, is called Plateaus Problem. Trans. 



394 THE SCIENCE OF MECHANICS. 

one another at the edges of the framework. When 
thin plane films are so joined that they meet at a hol- 
low edge, the law 1/r -{- ]//-' = const no longer holds 
for the liquid surface, as this sum has the value zero 
for plane surfaces and for the hollow edge a very large 
negative value. Conformably, therefore, to the views 
above reached, the liquid should run out of the films, 
the thickness of which would constantly decrease, and 
escape at the edges. This is, in fact, what happens. 
But when the thickness of the films has decreased to a 
certain point, then, for physical reasons, which are, as 
it appears, not yet perfectly known, a state of equilib- 
rium is effected. 

Yet, notwithstanding the fact that the fundamental 
equation l/r-\-l/r' = const is not fulfilled in these fig- 
ures, because very thin liquid films, especially films of 
viscous liquids, present physical conditions somewhat 
different from those on which our original suppositions 
were based, these figures present, nevertheless, in all 
cases a minimum of superficial area. The liquid films, 
connected with the wire edges and with one another, 
always meet at the edges by threes at approximately 
equal angles of 120 , and by fours in corners at approxi- 
mately equal angles. And it is geometrically demon- 
strable that these relations correspond to a minimum 
of superficial area. In the great diversity of phenom- 
ena here discussed but one fact is expressed, namely 
that the molecular forces do work, positive work, when 
the superficial area is diminished. 
The reason 7- The figures of equilibrium which Plateau ob- 
equiHbrium tained by dipping wire polyhedrons in solutions of 
metneai. soap, form systems of liquid films presenting a re- 
markable symmetry. The question accordingly forces 
itself upon us, What has equilibrium to do with sym- 



THE EXTEXSIOX OF THE PRIXCIPLES. 



395 



metry and regularity ? The explanation is obvious. 
In every symmetrical system every deformation that 
tends to destroy the symmetry is complemented by an 
equal and opposite deformation that tends to restore it. 
In each deformation positive or negative work is done. 
One condition, therefore, though not an absolutely 
sufficient one, that a maximum or minimum of work 
corresponds to the form of equilibrium, is thus sup- 
plied by symmetry. Regularity is successive symme- 
try. There is no reason, therefore, to be astonished 
that the forms of equilibrium are often symmetrical 
and regular. 

8. The science of mathematical hydrostatics arose The figure 

i -i 11 i r / oftheearth 

in connection with a special problem that of the figure 




Fig. 306. 

of the earth. Physical and astronomical data had led 
ton and Huygens to the view that the earth is an 
oblate ellipsoid of revolution. Newtox attempted to 
calculate this oblateness by conceiving the rotating 
earth as a fluid mass, and assuming that all fluid fila- 
ments drawn from the surface to the centre exert the 
same pressure on the centre. Huygens's assumption 
was that the directions of the forces are perpendicular 
to the superficial elements. Bouguer combined both 
assumptions. Clairaui, finally (Theorie de la figure 
de la terre, Paris, 1743), pointed out that the fulfilment 
of both conditions does not assure the subsistence of 
equilibrium. 



396 



THE SCIENCE OF MECHANICS. 



Clairaut's 
point of 
view. 



Conditions 
of equilib- 
rium of 
Clairaut's 
canals. 



.Clairaut's starting-point is this. If the fluid earth 
is in equilibrium, we may, without disturbing its equi- 
librium, imagine any portion of it solidified. Accord- 
ingly, let all of it be solidified but a canal AB, of any 
form. The liquid in this canal must also be in equilib- 
rium. But now the conditions which control equilib- 
rium are more easily investigated. If equilibrium exists 
in every imaginable eanal of this kind, then the entire 
mass will be in equilibrium. Incidentally Clairaut re- 
marks, that the Newtonian assumption is realised when 
the canal passes through the centre (illustrated in Fig. 
206, cut 2), and the Huygenian when the canal passes 
along the surface (Fig. 206, cut 3). 

But the kernel of the problem, according to Clai- 
raut, lies in a different view. In all imaginable canals, 

Z 








M 



X 



Fig. 207. Fig. 208. 

even in one which returns into itself, the fluid must be 
in equilibrium. Hence, if cross-sections be made at 
any two points M and N of the canal of Fig. 207, the 
two fluid columns MPN and MQN must exert on the 
surfaces of section at M and N equal pressures. The 
terminal pressure of a fluid column of any such canal 
cannot, therefore, depend on the length and the form 
of the fluid column, but must depend solely on the po- 
sition of its terminal points. 

Imagine in the fluid in question a canal MN of any 
form (Fig. 208) referred to a system of rectangular co- 



THE EXTENSION OF THE PRINCIPLES. 397 

ordinates. Let the fluid have the constant density p Mathemat- 

. . . ical expres- 

and let the force-components A, i , Z acting on unit of sionof 

. .. these con- 

mass of the fluid in the coordinate directions, be func- ditions. and 

tions of the coordinates x, y, z of this mass. Let the quent Ren- 
element of length of the canal be called ds, and let its tjon of 

liquid equi- 

projections on the axes be dx, dy, dz. The force-corn- libnum. 
ponents acting on unit of mass in the direction of the 
canal are then X(dx/ds), Y(dy/ds), Z(dz/ds). Let 
q be the cross-section ; then, the total force impelling 
the element of mass pods in the direction ds, is 

This force must be balanced by the increment of pres- 
sure through the element of length, and consequently 
must be put equal to q . dp. We obtain, accordingly, 
dp p {Xdx -|- Ydy -f- Zdz). The difference of pres- 
sure (/) between the two extremities M and N is found 
by integrating this expression from M to N. But as this 
difference is not dependent on the form of the canal 
but solely on the position of the extremities M and N, 
it follows that p {Xdx -f- Ydy -f- Zdz), or, the density 
being constant, Xdx + Ydy -f- Zdz, must be a com- 
plete differential. For this it is necessary that 

v _dU . r _dU 7 _dU 
dx dy dz 

where U is a function of coordinates. Hence, according 
to Clair aut, the general condition of liquid equilibrium is, 
that the liquid be controlled by forces which can be ex- 
pressed as the partial differential coefficients of one and 
the same function of coordinates. 

9. The Newtonian forces of gravity, and in fact all 
central forces, forces that masses exert in the direc- 
tions of their lines of junction and which are functions 



398 THE SCIENCE OF MECHANICS. 

character of the distances between these masses, possess this 
forces property. Under the action of forces of this character 
produce the equilibrium of fluids is possible. If we know U, 

equilibrium . _ . , 

we may replace the first equation by 

(dU dU ,dU.\ 

or 

dp = pdU and p = pC/-\- const. 

The totality of all the points for which U = const 
is a surface, a so-called level surface. For this surface 
also / = const. As all the force-relations, and, as we 
now see, all the pressure-relations, are determined by 
the nature of the function U, the pressure-relations, 
accordingly, supply a diagram of the force-relations, 
as was before remarked in page 98. 
ciairaufs In the theory of Clairaut, here presented, is con- 

germ of the tained, beyond all doubt, the idea that underlies the 
potential, doctrine of force-function or potential, which was after- 
wards developed with such splendid results by La- 
place, Poisson, Green, Gauss, and others. As soon 
as our attention has been directed to this property of 
certain forces, namely, that they can be expressed as 
derivatives of the same function U, it is at once recog- 
nised as a highly convenient and economical course to 
investigate in the place of the forces themselves the 
function U. 

If the equation 

dp = p (Xdx + Ydy -f Zdz) = pdU 

be examined, it will be seen that Xdx-\- Ydy -f- Zdz 
is the element of the work performed by the forces on 
unit of mass of the fluid in the displacement ds, whose 
projections are dx, dy, dz. Consequently, if we trans- 
port unit mass from a point for which U= C 1 to an- 



THE EXTEXSIOX OF THE PRIXCIPLES. 399 

other point, indifferently chosen, for which U = C, Character- 
or, more generally, from the surface U=C X to the force-func- 
surface (7= C 2 , we perform, no matter by what path 
the conveyance has been effected, the same amount of 
work. All the points of the first surface present, with 
respect to those of the second, the same difference of 
pressure ; the relation always being such, that 

Pi-Px =P(C 2 -C l ), 

where the quantities designated by the same indices 
belong to the same surface. 

10. Let us picture to ourselves a group of such character- 
very closely adjacent surfaces, of which every two sue- ievef, or 
cessive ones differ from each other by the same, very Ua^r-" 
small, amount of work required to transfer a mass from 
one to the other ; in other words, imagine the surfaces 
U= C, 17= C+ dC, C7= C+adC, and so forth. 

A mass moving on a level surface evidently per- 
forms no work. Hence, every component force in a 
direction tangential to the 
surface is = ; and the di- 
rection of the resultant 
force 'is even-where normal 
to the surface. If we call dn 
the element of the normal 
intercepted between two 
consecutive surfaces, and/ 
the force requisite to con- 
vey unit mass from the 
one surface to the other 
through this element, the 

work done is/, dn = d C. As ^C is by hypothesis every- 
where constant, the force /= dC/dn is inversely pro- 
portional to the distance between the surfaces consid- 




400 



THE SCIENCE OF MECHANICS. 



ered. If, therefore, the surfaces U are known, the 
directions of the forces are given by the elements of a 
system of curves everywhere at right angles to these 
surfaces, and the inverse distances between the sur- 
faces measure the magnitude of the forces. * These sur- 
faces and curves also confront us in the other depart- 
ments of physics. We meet them as equipotential 
surfaces and lines of force in electrostatics and mag- 
netism, as isothermal surfaces and lines of flow in the 
theory of the conduction of heat, and as equipotential 
surfaces and lines of flow in the treatment of electrical 
and liquid currents. 
illustration ii. We will now illustrate the fundamental idea of 

of Clai- _ 

ram's doc- Clairaut s doctrine by another, very simple example. 

trine by a . / ' . 

simple Imagine two mutually perpendicular planes to cut the 

example. . , , , . . ,f 

paper at right angles in the straight lines OX and OY 
(Fig. 210). We assume that a force-function exists 
. U= xy, where x andj are the distances from the 
two planes. The force-components parallel to OX and 
OYare then respectively 



X = 



dU 
dx 



y 



and 



dU 

Ty 






* The same conclusion may be reached as follows. Imagine a wr.ter pipe 
laid from New York to Key West, with its ends turning up vertically, and of 
glass. Let a quantity of water be poured into it, and when equilibrium is 
attained, let its height be marked on the glass at both ends. These two marks 
will be on one level surface. Now pour in a little more water and again mark 
the heights at both ends. The additional water in New York balances the 
additional water in Key West. The gravities of the two are equal. I'.ut their 
quantities are proportional to the vertical distances between the marks. 
Hence, the force of gravity on a fixed quantity of water is inversely as those 
vertical distances, that is, inversely as the distances between- consecutive 
level surfaces. Trans. 



THE EXTENSION OF THE PRIXCIPLES. 



401 



The level surfaces are cylindrical surfaces, whose 
generating lines are at right angles to the plane of the 
paper, and whose directrices, xy ==. const, are equi- 
lateral hyperbolas. The lines of force are obtained by 
turning the first mentioned system of curves through 
an angle of 45 in the plane of the paper about 0. If 
a unit of mass pass 
from the point r to O 
by the route rpO, or 
rqO, or by any other 
route, the work done 
is always Op X Oq. 
If we imagine a 
closed canal OprqO 
filled with a liquid, 
the liquid in the ca- 
nal will be in equi- 
librium. If transverse 
sections be made at 
any two points, each 
section will sustain 
pressure. 

We will now modify the example slightly. Let the a modific - 
forces be X = y, Y= a, where a has a constant example, 
value. There exists now no function U so constituted 
that X = dU/dx and Y= dUjdy ; for in such a case it 
would be necessary that dXjdy = dY/dx, which is ob- 
viously not true. There is therefore no force-function, 
and consequently no level surfaces. If unit of mass 
be transported from r to O by the way of /, the work 
done is a X Oq. If the transportation be effected by 
the route rqO, the work done is a X Oq + Op X Oq. 
If the canal OprqO were filled with a liquid, the liquid 
could not be in equilibrium, but would be forced to 




Fig. 210. 



at both its surfaces the same 



4 02 THE SCIENCE OF MECHANICS. 

rotate constantly in the direction prqO. Currents of 
this character, which revert into themselves but con- 
tinue their motion indefinitely, strike us as something 
quite foreign to our experience. Our attention, how- 
ever, is directed by this to an important property of 
the forces of nature, to the property, namely, that the 
work of such forces may be expressed as a function of 
coordinates. Whenever exceptions to this principle 
are observed, we are disposed to regard them as appa- 
rent, and seek to clear up the difficulties involved. 
Torrice/U's i2. We shall now examine a few problems of liquid 

researches . 

on the veio- motion. The founder of the theory of hydrodynamics is 
qmd efflux. Torricelli. Torricelli,* by observations on liquids dis- 
charged through orifices in the bottom of vessels, dis- 
covered the following law. If the time occupied in the 
complete discharge of a vessel be divided into n equal 
intervals, and the quantity discharged in the last, the 
// th , interval be taken as the unit, there will be dis- 
charged in the {n 1 ) th , the {n 2) th , the (n 3) th 

interval, respectively, the quantities 3, 5, 7 . . . . and 
so forth. An analogy between the motion of falling 
bodies and the motion of liquids is thus clearly sug- 
gested. Further, the perception is an immediate one, 
that the most curious consequences would ensue if the 
liquid, by its reversed velocity of efflux, could rise 
higher than its original level. Torricelli remarked, 
in fact, that it can rise at the utmost to this height, 
and assumed that it would rise exactly as high if all 
resistances could be removed. Hence, neglecting all 
resistances, the velocity of efflux, v, of a liquid dis- 
charged through an orifice in the bottom of a vessel is 
connected with the height // of the surface of the liquid 
by the equation v = Vzgh ; that is to say, the velocity 

* De Motu Cravium Projector ni, 1643. 



THE EXTENSION OF THE PRINCIPLES. 403 

of efflux is the final velocity of a body freely falling 
through the height h, or liquid-head ; for only with 
this velocity can the liquid just rise again to the sur- 
face. * 

Torricelli's theorem consorts excellently with the Varignon's 

deduction 

rest of our knowledge of natural processes ; but we of the vei- 

feel, nevertheless, the need of a more exact insight, efflux. 

Yarignon attempted to deduce the principle from the 

relation between force and the momentum generated by 

force. The familiar equation pt = mv gives, if by a 

we designate the area of the basal orifice, by h the 

pressure-head of the liquid, by s its specific gravity, 

the acceleration of a freely falling body, by v the 

velocity of efflux, and by r a small interval of time, 

this result 

l arts 

g 

Here a lis represents the pressure acting during the 

time t on the liquid mass avxs/g. Remembering that 

v is a final velocity, we get, more exactly, 

v 
a . ts 
ahs . t 1 .v. 



g 

and thence the correct formula 

v 2 = % 2gh. 
13. Daniel Bernoulli investigated the motions of 
fluids by the principle of vis viva. We w?'ll now treat 
the preceding case from this point of view, only ren- 
dering the idea more modern. The equation which we 
employ is ps = ttiv- ii. In a vessel of transverse sec- 
tion q v Fig. 211), into which a liquid of the specific 

* The early inquirers deduce cheir propositions in the incomplete form of 
proportions, and therefore usually put v proportional to J g k or * A. 



4 o 4 



THE SCIENCE OF MECHANICS. 



Daniel Ber- gravity s is poured till the head // is reached, the surface 

noulli's . ...... 

treatment sinks, say, the small distance ////, and the liquid mass 

of the same .... , . . . . 

problem, q . dh . s/g is discharged with the velocity v. The work 

done is the same as though the weight q. dh. s had 

descended the distance h. The path of the motion in 

the vessel is not of consequence here. It makes no 

difference whether the stratum q . dh 

is discharged directly through the 

basal orifice, or passes, say, to a 

position a, while the liquid at a is 

displaced to b, that at b displaced to 

c, and that at c discharged. The work 

done is in each case q . dh . s . h. 

Equating this work to the vis viva of the discharged 




Fig. 211. 



liquid, we get 
9 



dh 



h = 



q . dh . s v 2 



A r 



or 



v = \/2gh. 



The sole assumption of this argument is that all 
the work done in the vessel appears as vis viva in the 
liquid discharged, that is to say, that the velocities 
within the vessel and the work spent in overcoming 
friction therein may be tieglected. This assumption is 
not very far from the truth if vessels of sufficient width 
are employed, and no violent rotatory motion is set up. 

Let us neglect the gravity of the liquid in the ves- 
whe'n pro- ux sel, and imagine it loaded by a movable piston, on 
whose surface-unit the pressure / falls. If the piston 
be displaced a distance dh, the liquid volume q . dh 
will be discharged. Denoting the density of the liquid 
by p and its velocity by v, we then shall have 

2/ 



duced by 
the pres- 
sure of 
pistons. 



q . p . dh = q . dh . p -^, or v 







THE EXTEXSIOX OF THE PRINCIPLES. 



405 



Wherefore, under the same pressure, different liquids 
are discharged with velocities inversely proportional to 
the square root of their density. It is generally sup- 
posed that this theorem is directly applicable to gases. 
Its form, indeed, is correct ; but the deduction fre- 
quently employed involves an error, which we shall 
now expose. 

14. Two vessels (Fig. 212) of equal cross-sections 
are placed side by side and connected with each other 
by a small aperture in the base of their dividing walls. 
For the velocity of flow through this aperture we ob- 
tain, under the same suppositions as before, 



The appli- 
cation of 
this last re- 
sult to the 
flow of 
gases. 



9 



dh . s z 



*i - h 2) = 9 f- .T,orv = V tg{k x *,> 



g 



If we neglect the gravity of the liquid and imagine 
the pressures / x and /> 2 produced by pistons, we shall 
similarly have v = I 2(/, / a )/P- For example, if the 
pistons employed be loaded with the weights P and 

. the weight P will sink the distance /; and Pji 
will rise the distance //. The work (P/2)/i is thus left, 
to generate the vis viva of the effluent fluid. 

A gas under such circumstances would behave dif- 
ferently. Supposing the gas to flow from the vessel 
containing the load Pinto that contain- 
ing the load Pji, the first weight will 
fall a distance h, the second, however, 
since under half the pressure a gas dou- 
bles its volume, will rise a distance 2/1, 
so that the work Ph {P/i) ih = to 
would be performed. In the case of 
gases, accordingly, some additional 
work, competent to produce the flow between the vessels 
must be performed. This work the gas itself performs, 
by expanding, and by overcoming by its force of expan- 



4 MW dh 



The behav- 
iour of a 
gas under 
the as- 
sumed con- 
ditions. 



Fig. 212. 



4 o6 THE SCIENCE OF MECHANICS. 

The result sion a pressure. The expansive force / and the volume 

the same in x 

form but w of a gas stand to each other in the familiar relation 

different in 

iM.i K nitude./z/ = k, where k, so long as the temperature of the 
gas remains unchanged, is a constant. Supposing the 
volume of the gas to expand under the pressure / by 
an amount dw, the work done is 



f* dw =*f-jr 



For an expansion from w to w, or for an increase of 
pressure from p to p, we get for the work 

o,)-og(* 

Conceiving by this work a volume of gas w Q of 
density p, moved with the velocity v, we obtain 

/; 



v /2/ log (7} 



P 

The velocity of efflux is, accordingly, in this case also 
inversely proportional to the square root of the density; 
Its magnitude, however, is not the same as in the case 
of a liquid, 
incom- But even this last view is very defective. Rapid 

plctcncss of 

this view, changes of the volumes of gases are always accom- 
panied with changes of temperature, and, consequently 
also with changes of expansive force. For this reason, 
questions concerning the motion of gases cannot be 
dealt with as questions of pure mechanics, but always 
involve questions of heat. [Nor can even a thermo- 
dynamical treatment always suffice : it is sometimes 
necessary to go back to the consideration of molecular 
motions.] 

15. The knowledge that a compressed gas contains 
stored- up work, naturally suggests the inquiry, whether. 



THE EXTENSION OF THE PRINCIPLES. 



407 



this is not also true of compressed liquids. As a mat- Relative 

volumes of 

ter of fact, every liquid under pressure is compressed, compressed 

. . . 1 > 1 gases and 

To effect compression work is requisite, which reap- liquids, 
pears the moment the liquid expands. But this work, 
in the case of the mobile liquids, is very small. Imag- 
ine, in Fig. 213, a gas and a mobile liquid of the same 
volume, measured by OA, subjected to the same pres- 
sure, a pressure of one atmosphere, designated b) r AB. 
If the pressure be reduced to one-half an atmosphere, 
the volume of the gas will be doubled, while that of 
the liquid will be increased by only about 25 millionths. 
The expansive work of the gas is represented by the 
surface ABDC, that of the liquid by ABLK, where 




Fig. 213. 

AK =0-0000250 A. If the pressure decrease till it 
become zero, the total work of the liquid is represented 
by the surface ABI, where AI = 000005OA, and the 
total work of the gas by the surface contained between 
AB, the infinite straight line ACEG . . . ., and the 
infinite hyperbola branch BDFH . . . . Ordinarily, 
therefore, the work of expansion of liquids may be 
neglected. There are however phenomena, for ex- 
ample, the soniferous vibrations of liquids, in which 
work of this very order plays a principal part. In such 
cases, the changes of temperature the liquids undergo 
must also be considered. We thus see that it is only 
by a fortunate concatenation of circumstances that we 
are at liberty to consider a phenomenon with any close 



408 



THE SCIENCE OF MECHANICS. 



The hydro- 
dynamic 
principle 
of Danii 1 
Bernoulli. 



approximation to the truth as a mere matter of molar 
mechanics. 

16. We now come to the idea which Daniel Ber- 
noulli sought to apply in his work Hydrodynamica, sive 
de Viribus et Motibus Fliiidorum Conuncntarii (1738). 
When a liquid sinks, the space through which its cen- 
tre of gravity actually descends {descensus actualis) is 
equal to the space through which the centre of gravity 
of the separated parts affected with the velocities ac- 
quired in the fall can ascend (ascensus potentialis'). This 
idea, we see at once, is identical with that employed 
by Huygens. Imagine a vessel filled with a liquid 
(Fig. 214); and let its horizontal cross- 
section at the distance x from the plane 
of the basal orifice, be called f(x). Let 
the liquid move and its surface descend 
a distance dx. The centre of gravity, 
then, descends the distance x/(x) . dx/M, 
where M= Cf(x) dx. If k is the space of 
potential ascent of the liquid in a cross- 
section equal to unity, the space of po- 
tential ascent in the cross-section f(x) will be. k/f(x )-, 
and the space of potential ascent of the centre of 
gravity will be 



/W 



\ 


^Jdx 






|r r : 


r ~l 


/r- ; 


_ . \ 






































\\ 


E=f 



Fig. 214. 






where 



M 



N: 



N 
'~M' 



J* dx 
A*) 



For the displacement of the liquid's surface through a 
distance dx, we get, by the principle assumed, both 
iVand k changing, the equation 

x/(x) dx = Ndk + kdN. 



THE EXTEXSIO.V OF THE PRIXCIPLES. 



409 



This equation was employed by Bernoulli in the solu- 
tion of various problems. It will be easily seen, that 
Bernoulli's principle can be employed with success 
only when the relative velocities of the single parts of 
the liquid are known. Bernoulli assumes, an assump- 
tion apparent in the formula?, that all particles once 
situated in a horizontal plane, continue their motion 
in a horizontal plane, and that the velocities in the 
different horizontal planes are to each other in the in- 
verse ratio of the sections of the planes. This is the 
assumption of the parallelism of strata. It does not, in 
many cases, agree with the facts, and in others its 
agreement is incidental. When the vessel as compared 
with the orifice of efflux is very wide, no assumption 
concerning the motions within the vessel is necessary, 
as we saw in the development of Torricelli's theorem. 
17. A few isolated cases of liquid motion were 
treated by Newton and John Bernoulli. We shall 
consider here one to which a 
familiar law is directly applic- 
able. A cylindrical U-tube with 
vertical branches is filled with 
a liquid (Fig. 215). The length 
of the entire liquid column is /. 
If in one of the branches the 
column be forced a distance x 
below the level, the column in 
the other branch will rise the distance x, and the 
difference of level corresponding to the excursion x 
will be 2 x. If a is the transverse section of the tube 
and s the liquid's specific gravity, the force brought 
into play when the excursion x is made, will be 2 asx, 
which, since it must move a mass a Is/g will determine 
the acceleration (2 asx)/(als/g) = (2^//) x, or, for unit 



The paral- 
lelism of 
strata. 



The water- 
pendulum 
of Newton. 



Fig. 115. 



4io THE SCIENCE OE MECHANICS. 

excursion, the acceleration ig/1. We perceive that 
pendulum vibrations of the duration 



*>ls 



T. 

*g 

will take place. The liquid column, accordingly, vi- 
brates the same as a simple pendulum of half the length 
of the column. 
The liquid A similar, but somewhat more general, problem was 

pendulum 

of John treated bv Tohn Bernoulli. The two branches of a 

Bernoulli. . , . , . , . . 

cylindrical tube (Fig. 216), curved in any manner, make 

with the horizon, at the 
points at which the 
surfaces of the liquid 
move, the angles a 
and /?. Displacing one 
of the surfaces the dis- 
tance x, the other sur- 
face suffers an equal 
displacement. A difference of level is thus produced 
x (sin a -|- sin/?), and we obtain, by a course of reason- 
ing similar to that of the preceding case, employing 
the same symbols, the formula 




Fig. 216. 



rp I 

\ g (sin a -\- sin ytf) ' 
The laws of the pendulum hold true exactly for the 
liquid pendulum of Fig. 215 (viscosity neglected), even 
for vibrations of great amplitude ; while for the filar 
pendulum' the law holds only approximately true for 
small excursions. 

18. The centre of gravity of a liquid as a whole can 
rise only as high as it would have to fall to produce its 
velocities. In every case in which this principle appears 
to present an exception, it can be shown that the excep- 



niE EXTEXSIOX OF THE PRINCIPLES. 



411 



n 



tion is only apparent. One example is Hero's fountain. 

This apparatus, as we know, consists of three vessels, 

which may be designated in the descending order as 

A, B, C. The water in the open vessel A 

falls through a tube into the closed vessel 

C; the air displaced in C exerts a pressure 

on the water in the closed vessel B, and 

this pressure forces the water in B in a 

jet above A whence it falls back to its 

original level. The water in B rises, it is 

true, considerably above the level of B, 

but in actuality it merely flows by the 

circuitous route of the fountain and the 

vessel A to the much lower level of C. 

Another ap- 
parent exception 
to the principle 
in question is 
that of Montgol- 
fier's hydraulic 
ram, in which the 
liquid by its own 
gravitational 
work appears to 
rise considerably 
above its original 
level. The liquid 
flows (Fig. 217) 
from a cistern A 
through a long 
pipe RR and a valve J', which opens inwards, into a 
vessel B. When the current becomes rapid enough, the 
valve V is forced shut, and a liquid mass m affected with 
the velocity v is suddenly arrested in RR, which must 



Hero's 
Et j:.:ain 




Fig. 217. 



4 I2 



THE SCIENCE OF MECHANICS. 



be deprived of its momentum. If this be done in the 
time /, the liquid can exert during this time a pressure 
q = tnv/t, to which must be added its hydrostatical 
pressure p. The liquid, therefore, will be able, during 
this interval of time, to penetrate with a pressure/ -(- q 
through a second valve into a pila Heronis, H, and in 
consequence of the circumstances there existing win 
rise to a higher level in the ascension-tube 61S* than 
that corresponding to its simple pressure /. It is 
to be observed here, that a considerable portion of the 
liquid must first flow off into B, before a velocity requi- 
site to close V\s produced by the liquid's work in RR. 
A small portion only rises above the original level ; 
the greater portion flows from A into B. If the liquid 
discharged from S.S were collected, it could be easily 
proved that the centre of gravity of the quantity thus 
discharged and of that received in B lay, as the result 
of various losses, actually below the level of A. 
An niustra- The principle of the hydraulic ram, that of the 

tion, which . 

elucidates transference of work done by a large liquid mass to a 

smaller one, which 
thus acquires a great 
vis viva, may be illus- 
trated in the following 
very simple manner. 
Close the narrow 
opening O of a funnel 
and plunge it, with its 
wide opening down- 
wards, deep into a 
large vessel of water. If the finger closing the upper 
opening be quickly removed, the space inside the 
funnel will rapidly fill with water, and the surface of the 
water outside the funnel will sink. The work performed 




Fig. 218. 



THE EXTEXSIOX OF THE PRINCIPLES. 413 

is equivalent to the descent of the contentg of the funnel 
from the centre of gravity S of the superficial stratum 
to the centre of gravity S' of the contents of the fun- 
nel. If the vessel is sufficiently wide the velocities in 
it are all very small, and almost the entire vis viva is 
concentrated in the contents of the funnel. If all the 
parts of the contents had the same velocities, they 
could all rise to the original level, or the mass as a 
whole could rise to the height at which its centre of 
gravity was coincident with S. But in the narrower 
sections of the funnel the velocity of the parts is 
greater than in the wider sections, and the former 
therefore contain by far the greater part of the vis 
viva. Consequently, the liquid parts above are vio- 
lently separated from the parts below and thrown 
out through the neck of the funnel high above the 
original surface. The remainder, however, are left 
considerably below that point, and the centre of grav- 
ity of the whole never as much as reaches the original 
level of S. 

19. One of the most important achievements of Hydrostatic 

t-v ......... . . , and hvdro- 

Damel Bernoulli is his distinction of hydrostatic and dynamic 

/ J j tu pressure. 

hyaroaynamic pressure. The pressure 

which liquids exert is altered by motion ; 

and the pressure of a liquid in motion 

may, according to the circumstances, be 

greater or less than that of the liquid at rest 

with the same arrangement of parts. We 

will illustrate this by a simple example. 

The vessel A, which has the form of a body 

of revolution with vertical axis, is kept 

constantly filled with a frictionless liquid, so that its 

surface at mn does not change during the discharge 

at kl. We will reckon the vertical distance of a particle 




4M THE SCIENCE OF MECHANICS. 

Determina- from the surface ;;/;/ downwards as positive and call 

tion of the . _ , 

pressures it z. Let us follow the course of a prismatic element of 
acting in li- volume, whose horizontal base-area is a and height 6, 

quids in . . . . 

motion. in its downward motion, neglecting, on the assump- 
tion of the parallelism of strata, all velocities at right 
angles to z. Let the density of the liquid be p, the 
velocity of the element v, and the pressure, which is 
dependent on z, p. If the particle descend the dis- 
tance dz, we have by the principle of vis viva 

a pd[*~A = a ft pgdz ot d tfidz (1) 

that is, the increase of the vis viva of the element is 
equal to the work of gravity for the displacement in 
question, less the work of the forces of pressure of the 
liquid. The pressure on the upper surface of the element 
is ap, that on the lower surface is a \_p -\- (dp/dz)fi~\. 
The element sustains, therefore, if the pressure in- 
crease downwards, an upward pressure a (dp/dz)fi ; 
and for any displacement dz of the element, the work 
a(dp/dz~)ftdz must be deducted. Reduced, equation 
(i) assumes the form 

p.d{^ = pgdz- d dz 

and, integrated, gives 

P- l ~>- = Pg Z J>+ COnSt ( 2 ) 

If we express the velocities in two different hori- 
zontal cross-sections a x and a 2 at the depths z x and s 2 
below the surface, by v x , v 2 , and the corresponding 
pressures by p x , p 2 , we may write equation (2) in the 
form 



THE EXTENSION OF THE PRIXCIPLES. 415 
Taking for our cross-section a. the surface, z. = 0, The hjdiu 

' . dvnamic 

/. = ii; and as the same quantity of liquid flows through pressure 

r ' . -ir- varies with 

all cross-sections in the same interval of time, a. v. = thecircnm- 

1 * stances of 

Whence, finally, the motion. 

A-P^ + f-tp^t) 

The pressure /, of the liquid in motion (the hydro- 
dynamic pressure) consists of the pressure pgz 2 of the 
liquid at rest (the hydrostatic pressure) and of a pres- 
sure (p/2)?'2[(a2 i)/"l] dependent on the density, 
the velocity of flow, and the cross-sectional areas. In 
cross-sections larger than the surface of the liquid, the 
hydrodynamic pressure is greater than the hydrostatic, 
and vice versa. 

A clearer idea of the significance of Bernoulli's niustration 

of these re- 

principle may be obtained by imagining the liquid in suits bv the 
the vessel A unacted on by gravitv, and its outflow quids under 

^ * pressures 

produced by a constant pressure / t on the surface, produced 
Equation (3) then takes the form 

If we follow the course of a particle thus moving, it 
will be found that to every increase of the velocity of 
flow (in the narrower cross-sections) a decrease of 
pressure corresponds, and to every decrease of the ve- 
locity of flow (in the wider cross-sections) an increase 
of pressure. This, indeed, is evident, wholly aside 
from mathematical considerations. In the present case 
every change of the velocity of a liquid element must be 
exclusively produced by the work of the liquid" s forces 
of pressure. When, therefore, an element enters into 
a narrower cross-section, in which a greater velocity 
of flow prevails, it can acquire this higher velocity only 



416 



THE SCIENCE OF MECHANICS. 



on the condition that a greater pressure acts on its rear 
surface than on its front surface, that is to say, only 
when it moves from points of higher to points of lower 
pressure, or when the pressure decreases in the direc- 
tion of the motion. If we imagine the pressures in 
a wide section and in a succeeding narrower section 
to be for a moment equal, the acceleration of the ele- 
ments in the narrower section will not take place ; the 
elements will not escape fast enough ; they will accumu- 
late before the narrower section ; and at the entrance 
to it the requisite augmentation of pressure will be im- 
mediately produced. The converse case is obvious. 
Treatment 20. In dealing with more complicated cases, the 

of a liquid .... . , . . 

problem in problems of liquid motion, even though viscosity be 

which vis- 
cosity and 
friction are 
considered.. 




Fig. 220. 

neglected, present great difficulties ; and when the 
enormous effects of viscosity are taken into account, 
anything like a dynamical solution of almost every 
problem is out of the question. So much so, that al- 
though these investigations were begun by Newton, 
we have, up to the present time, only been able to 
master a very few of the simplest problems of this class, 
and that but imperfectly. We shall content ourselves 
with a simple example. If we cause a liquid contained 
in a vessel of the pressure-head h to flow, not through 
an orifice in its base, but through a long cylindrical 
tube fixed in its side (Fig. 220), the velocity of efflux 



THE EXTEXSIOX OF THE PRIXCIPLES. 417 

v will be less than that deducible from Torricelli's law, 
as a portion of the work is consumed by resistances 
due to viscosity and perhaps to friction. We find, in 
fact, that v = V*gh xi where h x < h. Expressing by h x 
the velocity-head, and by h 2 the resistance-head, we may 
put h = h x -f- h 2 . If to the main cylindrical tube we 
affix vertical lateral tubes, the liquid will rise in the 
latter tubes to the heights at which it equilibrates the 
pressures in the main tube, and will thus indicate at all 
points the pressures of the main tube. The noticeable 
fact here is, that the liquid-height at the point of influx 
of the tube is = //,, and that it diminishes in the direc- 
tion of the point of outflow, by the law of a straight 
line, to zero. The elucidation of this phenomenon is 
the question now presented. 

Gravity here does not act directly on the liquid in The condi- 
the horizontal tube, but all effects are transmitted to it perform- 
by the pressure of the surrounding parts. If we imag-work in 
ine a prismatic liquid element of basal area a and 
length fi to be displaced in the direction of its length 
a distance dz, the work done, as in the previous case, is 

-a dp Bdz=-a(S dp -dz. 
dz H H dz 

For a finite displacement we have 
/i 

d dz = -afi( F2 -j> 1 ) (1) 

H 
Work is done when the element of volume is displaced 
from a place of higher to a place of lower pressure. 
The amount of the work done depends on the size of 
the element of volume and on the difference of pressure 
at the initial and terminal points of the motion, and 
not on the length and the form of the path traversed. 



-afij 



4i8 



THE SCIENCE OE MECHANICS. 



The conse- 
quences of 
these con- 
ditions. 



If the diminution ot pressure were twice as rapid in 
one case as in another, the difference of the pressures 
on the front and rear surfaces, or the force of the work, 
would be doubled, but the space through which the 
work was done would be halved. The work done would 
remain the same, whether done through the space ab 
or ac of Fig. 221. 

Through every cross-section q of the horizontal tube 
the liquid flows with the same velocity v. If, neglect- 
ing the differences of velocity in the same cross-section, 
we consider a liquid element which exactly fills the 
section q and has the length (5, the vis viva q ft p(v 2 /2) 
of such an element will persist unchanged throughout 
its entire course in the tube. 
This is possible only provided 
the vis viva consumed by friction 
is replaced by the work of the 
liquid'' s forces of pressure. Hence, 
in the direction of the motion 
of the element the pressure 
must diminish, and for equal distances, to which the 
same work of friction corresponds, by equal amounts. 
The total work of gravity on a liquid element qfip 
issuing from the vessel, is q ft pgh. Of this the portion 
q p p(v 2 /2) is the vis viva of the element discharged 
with the velocity v into the mouth of the tube, or, as 
v = V ' 7.gh x , the portion q (3 pgh x . The remainder of 
the work, therefore, q (3 pgh 2 , is consumed in the tube, 
if owing to the slowness of the motion W3 neglect the 
losses within the vessel. 

If the pressure-heads respectively obtaining in the 
vessel, at the mouth, and at the extremity of the tube, 
are h, h 2 , 0, or the pressures are/ = /igp, p 2 =- h 2 gp,$, 
then by equation (1) of page 417 the work requisite to 




Fig. 221. 



THE EXT EX SI OX OF THE PRIXCIPLES. 419 

generate the vis viva of the element discharged into 
the mouth of the tube is 

and the work transmitted by the pressure of the liquid 
to the element traversing the length of the tube, is 

qPPi = qPgphv 

or the exact amount consumed in the tube. 

Let us assume, for the sake of argument, that the indirect 

, , - , , demonstra- 

pressure does not decrease from/ 2 at the mouth totionof 
zero at the extremity of the tube by the law of a straight sequences, 
line, but that the distribution of the pressure is differ- 
ent, say, constant throughout the entire tube. The 
parts in advance then will at once suffer a loss of ve- 
locity from the friction, the parts which follow will 
crowd upon them, and there will thus be produced at 
the mouth of the tube an augmentation of pressure 
conditioning a constant velocity throughout its entire 
length. The pressure at the end of the tube can only 
be = because the liquid at that point is not prevented 
from yielding to any pressure impressed upon it. 

If we imagine the liquid to be a mass of smooth a simile 
elastic balls, the balls will be most compressed at the which these 
bottom of the vessel, they will enter the tube in a state maTbe"" 1 
of compression, and will gradually lose that state in cefvid. 
the course of their motion. We leave the further de- 
velopment of this simile to the reader. 

It is evident, from a previous remark, that the work 
stored up in the compression of the liquid itself, is very 
small. The motion of the liquid is due to the work of 
gravity in the vessel, which by means of the pressure 
of the compressed liquid is transmitted to the parts in 
the tube. 



420 



THE SCIENCE OF MECHANICS. 



a partial An interesting modification of the case just cas- 

eation of cussed is obtained by causing the liquid to flow through 
discussed, a tube composed of a number of shorter cylindrical 
tubes of varying widths. The pressure in the direction 
of outflow then diminishes (Fig. 222) more rapidly in 
the narrower tubes, in which a greater consumption of 
work by friction takes place, than in the wider ones. 
We further note, in every passage of the liquid into a 




Fig. 222. 

wider tube, that is to a smaller velocity of flow, an in- 
crease of pressure (a positive congestion) ; in every 
passage into a narrower tube, that is to a greater velo- 
city of flow, an abrupt diminution of pressure (a nega- 
tive congestion). The velocity of a liquid element on 
which no direct forces act can be diminished or in- 
creased only by its passing to points of higher or lower 
pressure. 



CHAPTER IV. 

THE FORMAL DEVELOPMENT OF MECHANICS. 

i. 

THE ISOPERIMETRICAL PROBLEMS. 

i. When the chief facts of a physical science have The forma], 

. . as distin- 

once been fixed by observation, a new period of its guished 

from the de- 
development begins the deductive, which we treated ductive, de- 

. velopment 

in the previous chapter. In this period, the facts are of physical 

science. 

reproducible in the mind without constant recourse to 
observation. Facts of a more general and complex 
character are mimicked in thought on the theory that 
they are made up of simpler and more familiar obser- 
vational elements. But even after we have deduced 
from our expressions for the most elementary facts 
(the principles) expressions for more common and more 
complex facts (the theorems) and have discovered in 
all phenomena the same elements, the developmental 
process of the science is not yet completed. The de- 
ductive development of the science is followed by its 
formal development. Here it is sought to put in a clear 
compendious form, or system, the facts to be repro- 
duced, so that each can be reached and mentally pic- 
tured with the least intellectual effort. Into our rules 
for the mental reconstruction of facts we strive to in- 
corporate the greatest possible uniformity, so that these 
rules shall be easy of acquisition. It is to be remarked, 
that the three periods distinguished are not sharply 



422 THE SCIENCE OE MECHANICS. 

separated from one another, but that the processes of 
development referred to frequently go hand in hand, 
although on the whole the order designated is unmis- 
takable. 
Theisoperi- 2. A powerful influence was exerted on the formal 

metrical 

problems, development of mechanics by a particular class of 

and ques- m * ' * 

tions of mathematical problems, which, at the close of the 

maxima 

ana minima seventeenth and the beginning of the eighteenth cen- 
turies, engaged the deepest attention of inquirers. 
These problems, the so-called isoperimetrical problems, 
will now form the subject of our remarks. Certain 
questions of the greatest and least values of quanti- 
ties, questions of maxima and minima, were treated by 

the Greek mathemati- 
cians. Pythagoras is 
said to have taught that 
the circle, of all plane 
* figures of a given pen- 

Fi meter, has the greatest 

area. The idea, too, of a 
certain economy in the processes of nature was not 
foreign to the ancients. Hero deduced the law of the 
reflection of light from the theory that light emitted 
from a point A (Fig. 223) and reflected at M will travel 
to B by the shortest route. Making the plane of the 
paper the plane of reflection, .SS the intersection of 
the reflecting surface, A the point of departure, B the 
point of arrival, and M the point of reflection of the 
ray of light, it will be seen at once that the line AMB', 
where B' is the reflection of B, is a straight line. The 
line AMB' is shorter than the line ANB ', and there- 
fore also AMB is shorter than ANB. Pappus held 
similar notions concerning organic nature ; he ex- 




FORMAL DEVELOPMEXT. 423 

plained, for example, the form of the cells of the honey- 
comb by the bees' efforts to economise in materials. 

These ideas fell, at the time of the revival of the The re- 
sciences, on not unfruitful soil. They were first taken Kepier.Fer- 
up by Fermat and Roberval, who developed a method Robervai. 
applicable to such problems. These inquirers ob- 
served, as Kepler had already done, that a magni- 
tude v which depends on another magnitude x, gen- 
erally possesses in the vicinity of its greatest and least 
values a peculiar property. Let x (Fig. 224) denote 
abscissas and v ordinates. If, while x increases, y pass 
through a maximum value, its increase, or rise, will 
be changed into a decrease, or 
fall ; and if it pass through a 
minimum value its fall will be 
changed into a rise. The neigh- 
boring values of the maximum 
or minimum value, consequently, Fig ^ 

will lie very near each other, and 

the tangents to the curve at the points in question will 
generally be parallel to the axis of abscissas. Hence, 
to find the maximum or minimum values of a quan- 
tity, we seek the parallel tangents of its curve. 

The method of tangents may be put in analytical The 
form. For example, it is required to cut off from a tangents, 
given line a a portion .r such that the product of the 
two segments x and a x shall be as great as possible. 
Here, the product x (a x) must be regarded as the 
quantity v dependent on x. At the maximum value of 
v any infinitely small variation of x, say a variation , 
will produce no change in y. Accordingly, the required 
value of x will be found, by putting 

x (a x) = (x + S) (a x g) 
or 




424 



THE SCIENCE OF MECHANICS. 



ax x 2 =ax -\- at; x 2 xB, x B 2 
or 

= a 2x $. 

As B, may be made as small as we please, we also get 

= a 2x; 
whence x = a/2. 

In this way, the concrete idea of the method of 
tangents may be translated into the language of alge- 
bra ; the procedure also contains, as we see, the germ 
of the differential calculus. 
The refrac- Fermat sought to find for the law of the refraction 

tion of light . . . . , . 

as a mini- of light an expression analogous to that of Hero for 

law of reflection. He remarked 
that light, proceeding from a 
point A, and refracted at a 
point M, travels to B, not by 
the shortest route, but in the 
shortest time. If the path AMB 
is performed in the shortest 
time, then a neighboring path 
ANB, infinitely near the real 
path, will be described in the 
same time. If we draw from N on AM and from M on 
NB the perpendiculars NP and MQ, then the second 
route, before refraction, is less than the first route by a 
distance MP= NM sin a, but is larger than it after 
refraction by the distance NQ = NM sin /3. On the 
supposition, therefore, that the velocities in the first 
and second media are respectively v j and v 2 , the time 
required for the path AMB will be a minimum when 

NM sin a NMsin ft 




Fig. 225. 



or 



FORMAL DEVELOPMENT. 425 

v x sinar 
v 2 sin// 

where n stands for the index of refraction. Hero's law 
of reflection, remarks Leibnj/fe, is thus a special case 
of the law of refraction. For equal velocities (<'j = r 2 \ 
the condition of a minimum of time is identical with 
the condition of a minimum of space. 

Huvgens, in his optical investigations, applied and Huygens's 

1 completion 

further perfected the ideas of Fermat, considering, not of Fermats 
only rectilinear, but also curvilinear motions of light, 
in media in which the velocity of the light varied con- 
tinuously from place to place. For these, also, he 
found that Fermat's law obtained. Accordingly, in all 
motions of light, an endeavor, so to speak, to produce 
results in a minimum of time appeared to be the funda- 
mental tendency. 

3. Similar maximal or minimal properties were The prob- 
brought out in the study of mechanical phenomena, brachisto- 

chrone. 

As we have already noticed, John Bernoulli knew that 
a freely suspended chain assumes the form for which 
its centre of gravity lies lowest. This idea was, of 
course, a simple one for the investigator who first rec- 
ognised the general import of the principle of virtual 
velocities. Stimulated by these observations, inquir- 
ers now began generally to investigate maximal and 
minimal characters. The movement received its most 
powerful impulse from a problem propounded by John 
Bernoulli, in June, 1696* the problem of the brachis- 
tochrone. In a vertical plane two points are situated, 
A and B. It is required to assign in this plane the 
cun e by which a falling body will travel from A to B 
in the shortest time. The problem was very ingeniously 

* Acta Eruditorum, Leipsic. 



426 THE SCIENCE CE MECHANICS. 

solved by John Bernoulli himself ; and solutions were 
also supplied by Leibniz, L'Hopital, Newton, and 
James Bernoulli. 
John Her- The most remarkable solution was John Ber- 

noulli's in- , J 

geniousso- noulli's own, This inquirer remarks that problems 

lutinnof the * 

problem of of this class have already been solved, not for the mo- 

thebrachis- . 

tochrone. tion of falling bodies, but for the motion of light. He 
accordingly imagines the motion of a falling body re- 
placed by the motion of 
~ a ray of light. (Comp. 
P- 379-) The two points 

A and B are supposed 

^=^ to be fixed in a medium 

in which the velocity of 
Fig. 226. ... . J 

light increases in the 

vertical downward direction by the same law as the 
velocity of a falling body. The medium is supposed 
to be constructed of horizontal layers of downwardly 
decreasing density, such that v = V / 2gh denotes the 
velocity of the light in any layer at the distance h be- 
low A. A ray of light which travels from A to B un- 
der such conditions will describe this distance in the 
shortest time, and simultaneously trace out the curve 
of quickest descent. 

Calling the angles made by the element of the 
curve with the perpendicular, or the normal of the 
layers, a, a', a". . . ., and the respective velocities 
v, v' , v" . . . ., we have 

sin at sin a' sin a" 

v v' v" 

or, designating the perpendicular distances below A 
by x, the horizontal distances from A by y, and the arc 
of the curve by s, 



FORMAL DEVELOPMENT. 437 







The bra- 

chi - 
chr 

=^ k ^^ 



whence follows 

4j* = k* r* ds* = k* r 3 (Jx* + </>*) 
and because v = V igx also 



This is the differential equation of a cycloid, or curve 
described by a point in the circumference of a circle of 
radius r = a/2 = i/\gk 2 , rolling on a straight line. 

To find the cycloid that passes through A and B, The eon- 

1 m stracuonof 

- to be noted that all cycloids, inasmuch as they are the cycloid 

- - between 

produced by similar con- t k v 

-, -7R 1 . . points. 

structions, are similar, A\ 
and that if generated by 
the rolling of circles on 
AD from the point A as 
origin, are also similarly 

... Fig. 87. 

situated with respect to 

the point A. Accordingly, we draw through AB a 
straight line, and construct any cycloid, cutting the 
straight line in B". The radius of the generating 
circle is, say, r'. Then the radius of the generating 
circle of the cycloid sought is r= r\ABjABr). 

This solution of John Bernoulli's, achieved entirely 
without a method, the outcome of pure geometrical 
fancy and a skilful use of such knowledge as happened 
to be at his command, is one of the most remarkable 
and beautiful performances in the history of physical 
science. John Bernoulli was an aesthetic genius in this 
field. His brother James's character was entirely differ- 
ent. James was the superior of John in critical power, 




428 



THE SCIENCE OF MECHANICS. 



Compari- but in originality and imagination was surpassed by the 

son of the ... 

scientific latter. James Bernoulli likewise solved this problem, 

characters , ... .. . , -r-> , , , 

of John and though in less felicitous form. But, on the other hand, 

lames Ber- ,. 

icmiii. he did not fail to develop, with great thoroughness, a 
. general method applicable to such problems. Thus, 
in these two brothers we find the two fundamental 
traits of high scientific talent separated from one 
another, traits, which in the very greatest natural 
inquirers, in Newton, for example, are combined to- 
gether. We shall soon see those two tendencies, which 
within one bosom might have fought their battles un- 
noticed, clashing in open conflict, in the persons of 
these two brothers. 



- 3&K 




Vignette to Leibniizii et yohannis Rernoullii comercium epistolicui, 
Lausanne and Geneva, Bousquet, 1745. 



James Ber- 4. James Bernoulli finds that the chief object of 
marks-on research hitherto had been to find the values of a vari- 

the Ecncrfll 

nature of able quantity, for which a second variable quantity, 

the new . . , . _ - 

problem, which is a function of the first, assumes its greatest or 
its least value. The present problem, however, is to find 



FORMAL DEVELOPMEXT. 429 

from among an infinite number of curves one which pos- 
sesses a certain maximal or minimal property. This, as 
he correctly remarks, is a problem of an entirely dif- 
ferent character from the other and demands a new 
method. 

The principles that Tames Bernoulli employed inTheprinci- 

1 r J pies em- 

the solution of this problem (Acta Eruditorum, May, ployed in 

James Ber- 

1607")* are as follows : nouiiisso- 

** ' . . lution. 

(1) If a curve has a certain property of maximum 
or minimum, every portion or element of the curve has 
the same property. 

(2) Just as the infinitely adjacent values of the 
maxima or minima of a quantity in the ordinary prob- 
lems, for infinitely small changes of the independent 
variables, are constant, so also is the quantity here to 
be made a maximum or minimum for the curve sought, 
for infinitely contiguous curves, constant. 

(3) It is finally assumed, for the case of the brachis- 
tochrone, that the velocity is p =.\Zig h, where // de- 
notes the height fallen through. 

If we picture to ourselves a very small portion ABCrheessea- 
of the curve (Fig. 228), and, imagining a horizontal tures of 
line drawn through B, cause nouiii'sso- 

. A\ lution. 

the portion taken to pass into 
the infinitely contiguous por- 
tion ADC, we shall obtain, by 
considerations exactly similar 
to those employed in the treat- 
ment of Fermat's law, the well- 
Fig. 228. 
known relation between the 

sines of the angles made by the curve-elements with 
the perpendicular and the velocities of descent. In 
this deduction the following assumptions are made, 

* See also his works. Vol. II, p. 768. 




430 



THE SCIENCE OF MECHANICS. 



(i), that the part, or element, ABC is brachistochro- 
nous, and (2), that ADC is described in the same time 
as ABC. Bernoulli's calculation is very prolix ; but 
its essential features are obvious, and the problem is 
solved * by the above-stated principles. 
The Pro- With the solution of the problem of the brachisto- 

gramma of 

james Ber- chrone, Tames Bernoulli, in accordance with the prac- 

noulh, or . J . * 

theproposi-tice then prevailing among mathematicians, proposed 
general iso- the following more general " isoperimetrical problem ": 

perimetri- .... . 

cai prob- < Of all isoperimetrical curves (that is, curves of equal 
"perimeters or equal lengths) between the same two 
"fixed points, to find the curve such that the space 
"included (1) by a second curve, each of whose ordi- 
" nates is a given function of the corresponding ordi- 
"nate or the corresponding arc of the one sought, (2) 
"by the ordinates of its extreme points, and (3) by the 
" part of the axis of abscissas lying between those ordi- 
" nates, shall be a maximum or minimum." 

For example. It is required to find the curve BFN, 
described on the base BN such, that of all curves of 

the same length on BN, 
this particular one shall make 
the area BZN a minimum, 
where PZ = (PF)", LM=-- 
(LK)", and so on. Let the 
relation between the ordi- 
nates of BZN and the cor- 
responding ordinates of BFN 
be given by the curve BH. To obtain PZ from PF, 
draw FGH at right angles to BG, where BG is at right 
angles to BN. By hypothesis, then, PZ= GH, and 




Fig. 229. 



* For the details of this solution and for information generally on the his- 
tory of this subject, see Woodhouse's Treatise on Isoperimetrical Problems 
and the Calculus of Variations, Cambridge, 1810. Trans. 



FORMAL DEVELOPMENT. 431 

so for the other ordinates. Further, we put BP = y, 

FF= x, PZ = x m . 

John Bernoulli gave, forthwith, a solution of this John Ber- 
. nouiii's so- 

problem, in the form lutionof 

this prob- 
x H dx lem - 






where a is an arbitrary constant. For n = 1, 

/* x d x / 

y=i =a Va'x*, 

J \ a 2 x- 

that is, BFN is a semicircle on BN as diameter, and 
the area BZN is equal to the area BFN. For this par- 
ticular case, the solution, in fact, is correct. But the 
general formula is not universally valid. 

On the publication of John Bernoulli's solution, 
James Bernoulli openly engaged to do three things : 
fiFst, to discover his brother's method ; second, to point 
out its contradictions and errors ; and, third, to give the 
true solution. The jealousy and animosity of the two 
brothers culminated, on this occasion, in a violent and 
acrimonious controversy, which lasted till James's 
death. After James's death, John virtually confessed 
his error and adopted the correct method of his brother. 

James Bernoulli surmised, and in all probability James Ber- 

. , _, -tii 1 nouiii's 

correctly, that John, misled by the results of his re- criticism of 
searches on the catenary and the curve of a sail filled nouiii's so- 
with wind, had again attempted an indirect solution, 
imagining BFN filled with a liquid of variable density 
and taking the lowest position of the centre of gravity 
as determinative of the curve required. Making the 
ordinate PZ=p, the specific gravity of the liquid in 
the ordinate PF = x must be p/x, and similarly in 
every other ordinate. The weight of a vertical fila- 



432 



THE SCIENCE OE MECHANICS. 



ment is then / . dy/x, and its moment with respect to 
BNis 

1 / dy 1 , 

Hence, for the lowest position of the centre of gravity, 
i fPdy> or Cpdy =.BZN, is a maximum. But the 
fact is here overlooked, remarks James Bernoulli, that 
with the variation of the curve BFN the weight of the 
liquid also is varied. Consequently, in this simple 
form the deduction is not admissible. 
Thefunda- In the solution which he himself gives, Tames Ber- 

mental , J 

principle of noulli once more assumes that the small portion F F,,, 

James Ber- 

of the curve possesses the prop- 
erty which the whole curve pos- 
sesses. And then taking the four 
successive points F F, F fl F nn 
of which the two extreme ones 
are fixed, he so varies F, and 
F,, that the length of the arc F 
F, F n F, n remains unchanged, 
which is possible, of course, only by a displacement 
of two points. We shall not follow his involved and 
unwieldy calculations. The principle of the process is 
clearly indicated in our remarks. Retaining the des- 
ignations above employed, James Bernoulli, in sub- 
stance, states that when 

p dx 

d >' = /% =r> 

Va 2 J> 2 

Cpdy is a maximum, and when 

(a f) dx 




Fig. 230. 



dy = 



V'lapp 2 



S*'y 



is a minimum. 



FORMAL DEVELOPMENT. 433 

The dissensions between the two brothers were, we 
may admit, greatly to be deplored. Yet the genius of 
the one and the profundity of the other have borne, in 
the stimulus which Euler and Lagrange received from 
their several investigations, splendid fruits. 

5. Euler (Problematis Isoperimetrici Solutio Generalise Euier-s 

general 

Com. Acad. Petr. T. VI, for 1733, published in i738)*ciassinca- 

I , 1 , t'onofthe 

was the first to give a more general method of treating isoperimet- 

... ... rical prob- 

these questions of maxima and minima, or isoperimetri- lems. 
cal problems. But even his results were based on 
prolix geometrical considerations, and not possessed of 
analytical generality. Euler divides problems of this 
category, with a clear perception and grasp of their 
differences, into the following classes : 

(1) Required, of all curves, that for which a prop- 
erty A is a maximum or minimum. 

(2) Required, of all curves, equally possessing a 
property A, that for which B is a maximum or mini- 
mum. 

Required, of all curves, equally possessing two 
properties, A and B, that for which C is a maximum 
or minimum. And so on. 

A problem of the first class is (Fig. 231) the finding Example* 
of the shortest curve through M and N. A problem of 
the second class is the finding of a curve through M 
and N t which, having the given length A, makes the 
area J//Wa maximum. A problem of the third class 
would be : of all curves of the given length A, which 
pass through M t N and contain the same area 
MPN=B, to find one which describes when rotated 
about MN the least surface of revolution. And so on. 

* Euler's principal contributions to this subject are contained in three 
memoirs, published in the Commentaries of Petersburg for the years 1733, 1736, 
and 1766, and in the tract Methodus inz'eniendi Lineas Curvas Proprietate 
Minimive gaudentes, Lausanne and Geneva, 1744. Trams. 





434 THE SCIEXCE OF MECHANICS. 

We may observe here, that the finding of an abso- 
lute maximum or minimum, without collateral condi- 
tions, is meaningless. Thus, all the curves of which in 
the first example the shortest is sought 
possess the common property of pas- 
sing through the points Af and N. 

The solution of problems of the 
first class requires the variation of two 
elements of the curve or of one point. 
This is also sufficient. In problems 
of the second class three elements or 
Fig. 231. iwo points must be varied ; the reason 

being, that the varied portion must 
possess in common with the unvaried portion the prop- 
erty A, and, as B is to be made a maximum or mini- 
mum, also the property B, that is, must satisfy two con- 
ditions. Similarly, the solution of problems of the third 
class requires the variation of four elements. And 
so on. 
The com- The solution of a problem of a higher class involves, 

mutability ... . . . 

of theiso- by implication, the solution of its converse, in all its 

perimetri- ..,.,, 

cai proper- forms. Thus, in the third class, we vary four elements 

ties, with . . 

Rulers in- of the curve, so, that the varied portion of the curve 

ferences. . . . 

shall share equally with the original portion the values 
A and B and, as C is to be made a maximum or a 
minimum, also the value C. But the same conditions 
must be satisfied, if of all curves possessing equally B 
and C that for which A is a maximum or minimum is 
sought, or of all curves possessing A and C that for 
which B is a maximum or minimum is sought. Thus 
a circle, to take an example from the second class, con- 
tains, of all lines of the same length A, the greatest ! 
area B, and the circle, also, of all curves containing 
the same area B, has the shortest length A. As the 



FORMAL DEVELOPMEXT. 435 

condition that the property A shall be possessed in 
common or shall be a maximum, is expressed in the 
same manner, Euler saw the possibility of reducing the 
problems of the higher classes to problems of the first 
class. If, for example, it is required to find, of all 
curves having the common property A, that which 
makes B a maximum, the curve is sought for which 
mB is a maximum, where m is an arbitrary con- 
stant. If on any change of the curve, A -f- mB, for any 
value of m, does not change, this is generally possible 
only provided the change of A, considered by itself, 
and that of B, considered by itself, are = 0. 

6. Euler was the originator of still another impor- The funda- 
tant advance. In treating the problem of finding the principle of 
brachistochrone in a resisting medium, which was in- norm's er 
vestigated by Herrmann and him, the existing meth- nown not 
ods proved incompetent. For the brachistochrone in lei-Mii"'" 
a vacuum, the velocity depends solely on the vertical '" 
height fallen through. The velocity in one portion of 
the curve is in no wise dependent on the other por- 
tions. In this case, then, we can indeed say, that if 
the whole curve is brachistochronous, every element 
of it is also brachistochronous. But in a resisting 
medium the case is different. The entire length and 
form of the preceding path enters into the determina- 
tion of the velocity in the element. The whole curve 
can be brachistochronous without the separate ele- 
ments necessarily exhibiting this property. By con- 
siderations of this character, Euler perceived, that the 
principle introduced by James Bernoulli did not hold 
universally good, but that in cases of the kind referred 
to, a more detailed treatment was required. 

7- The methodical arrangement and the great num- 
ber of the problems solved, gradually led Euler to sub- 



UtlnJl:-.. 



436 THE SCIENCE OF MECHANICS. 

Lagrange's stantially the same methods that Lagrange afterwards 

place in the . 

history of developed in a somewhat different form, and which 
lus of vari- now go by the name of the Calculus of Variations. First, 
John Bernoulli lighted on an accidental solution of a 
problem, by analogy. James Bernoulli developed, for 
the solution of such problems, a geometrical method. 
Euler generalised the problems and the geometrical 
method. And finally, Lagrange, entirely emancipating 
himself from the consideration of geometrical figures, 
gave an analytical method. Lagrange remarked, that 
the increments which functions receive in consequence 
of a change in their form are quite analogous to the in- 
crements they receive in consequence of a change of 
their independent variables. To distinguish the two 
species of increments, Lagrange denoted the former 
by S, the latter by d. By the observation of this anal- 
ogy Lagrange was enabled to write down at once the 
equations which solve problems of maxima and minima. 
Of this idea, which has proved itself a very fertile one, 
Lagrange never gave a verification ; in fact, did not 
even attempt it. His achievement is in every respect 
a peculiar one. He saw, with great economical in- 
sight, the foundations which in his judgment were suf- 
ficiently secure and serviceable to build upon. But 
the acceptance of these fundamental principles them- 
selves was vindicated only by its results. Instead of 
employing himself on the demonstration of these prin- 
ciples, he showed with what success they could be em- 
ployed. (Essai d'une nouvelle mcthode pour determiner 
les maxima et minima des formules integrates indefinies. 
Misc. Taur. 1762.) 

The difficulty which Lagrange's contemporaries and 
successors experienced in clearly grasping his idea, is 
quite intelligible. Euler sought in vain to clear up the 



FORMAL DEVELOPMENT. 437 

difference between a variation and a differential byThe mis- 
imagining constants contained in the function, with tions of La- 
the change of which the form of the function changed, idea. 
The increments of the value of the function arising 
from the increments of these constants were regarded 
by him as the variations, while the increments of the 
function springing from the increments of the indepen- 
dent variables were the differentials. The conception 
of the Calculus of Variations that springs from such a 
view is singularly timid, narrow, and illogical, and does 
not compare with that of Lagrange. Even Lindelof's 
modern work, so excellent in other respects, is marred 
by this defect. The first really competent presenta- 
tion of Lagrange's idea is, in our opinion, that of Jel- 
lett. * Jellett appears to have said what Lagrange per- 
haps was unable fully to say, perhaps did not deem it 
necessary to say. 

8. lellett's view is, in substance, this. Quantities Miens ex- 

J .... position of 

generally are divisible into constant and variable quan- the princi- 
ples of the 
titles ; the latter being subdivided into independent calculus of 

, , . Variations. 

and dependent variables, or such as may be arbitrarily 
changed, and such whose change depends on the 
change of other, independent, variables, in some way 
connected with them. The latter are called functions 
of the former, and the nature of the relation that con- 
nects them is termed the form of the function. Now, 
quite analogous to this division of quantities into con- 
stant and variable, is the division of the forms of func- 
tions into determinate (constant) and indeterminate (vari- 
able). If the form of a function, y = (p(x), is inde- 
terminate, or variable, the value of the function y can 
change in two ways : (1) by an increment dx of the 

* An Elementary Treatise on the Calculus of Variations. By the Rev. 
John Hewitt Jellett. Dublin, 185a 



438 THE SCIENCE OF MECHANICS. 

independent variable x, or (2) by a change oi/orm, by 
a passage from cp to cp x . The first change is the dif- 
ferential dy, the second, the variation 6y. Accord- 
ingly, 

dy = cp (x -f dx) cp (x), and 

S 3'= 9x (x)-(p(x). 

The object The change of value of an indeterminate function 

of the cal- 

cuius of va- due to a mere change of form involves no problem. 

nations ll- . r 

lustrated. just as the change of value of an independent variable 
involves none. We may assume any change of form 
we please, and so produce any change of value we 
please. A problem is not presented till the change in 
value of a determinate function (F) of an indetermi- 
nate function cp, due to a change of form of the included 
indeterminate function, is required. For example, if 
we have a plane curve of the indeterminate form y= 
<p{x), the length of its arc between the abscissae ^ 
and x, is 



*i 



-*"o -*"o 

a determinate function of an indeterminate function. 
The moment a definite form of curve is fixed upon, the 
value of 6" can be given. For any change of form of 
the curve, the change in value of the length of the arc, 
6S, is determinable. In the example given, the func- 
tion S does not contain the function y directly, but 
through its first differential coefficient dy/dx, which is 
itself dependent on y. Let u = F(y) be a determinate 
function of an indeterminate function y = cp (r) ; then 

6u =F{y + 6y) - F{y) = ~p Sy. 

Again, let u=F(y, dy/dx) be a determinate function 



FORMAL DEVELOPMENT. 439 

of an indeterminate function, y = <p(x). For a change 
of form of (p, the value of y changes by 6y and the 
value of dy/dx by 6(dy/dx). The corresponding change 
in the value of // is 

dy ' ' //y d x 

dx 

dv 

The expression 6 r is obtained by our definition from Expres- 

a X sions for 

the varia- 

<T = rf(j+* _ <h = J*J feren,ill dif - 

</ A - j x x J x - coefficients. 

Similarly, the following results are found : 

d 2 v d~ dy d 3 y d 3 dy 
die* = ~dx*~' dx 3 = 7x f 

and so forth. 

We now proceed to a problem, namely, the de- a problem, 
termination of the form of the function y = <p(x) that 
will render 



where 



U= , 



l dy d- v 



a maximum or minimum ; cp denoting an indetermi- 
nate, and Fa. determinate function. The value of U 
may be varied (1) by a change of the limits, x , x t . 
Outside of the limits, the change of the independent 
variables x, as such, does not affect U; accordingly, 
if we regard the limits as fixed, this is the only respect 
m which we need attend to x. The only other way 
(2) in which the value of 7 is susceptible of variation 



440 THE SCIENCE OE MECHANICS. 

is by a change of the form of y = q>(x)- This produces 
a change of value in 

y> "t~ j o > 
dx az 1 

amounting to 

and so forth. The total change in U, which we shall 
call DU, and to express the maximum-minimum con- 
dition put =0, consists of the differential dC/ and the 
variation 6(7. Accordingly, 

DU=dU + 6U=0. 
Expression Denoting by V x dx x and V Q dx Q the increments of 
variation of U due to the change of the limits, we then have 

the func- 
tion in *\ 

question. DU= V x dx 1 V Q dx Q + 6 J Vdx 3= 



V x dx x V dx +J8V. dx 



II. 



But by the principles stated on page 439 we further get 

*v dV * \ dV * d y . dV * d% y.L 

dy j d y dx j d y dx 

dx dx 2 

dV , , dVddy , dV d*dy , 

s y + -jt. TZT + 



dy dy dx ,d 2 y dx 2 

dx dx 2 

For the sake of brevity we put 

d Z- N d Z- F r= F 
d y ~" JUL ~~ d-i 

dx dx 2 

Then 

S*fVdx = 



FORMAL DEVELOPMEXT. 4^1 

-^o ' the third 

term of the 

One difficulty here is, that not only 6 v, but also the ?*pressn 

* for the total 

terms d6y/dx, d- 6y/dx 2 .... occur in this equation, ^nation. 
terms which are dependent on one another, but not 
in a directly obvious manner. This drawback can be 
removed by successive integration by parts, by means 
of the formula 

Cudv = u v Cvd u. 
By this method 

d6y dP C d%P *x J a 

* -. . o y -\- I * oyax, and so on. 

Performing all these integrations between the limits, 
we obtain for the condition DC7=0 the expression 
"=l\dx x -V Q dx Q 

dP 2 

+ ('*-s + -.),k-('-^+-).. 

+ : 

which now contains only tf r under the integral sign. 

The terms in the first line of this expression are 
independent of any change in the form of the function 
and depend solely upon the variation of the limits. 



+ (A -'% + )*-('.- K...)* 



442 THE SCIENCE OF MECHANICS. 

The inter- The terms of the two following lines depend on the 

pretationof , . . ,,... 

the results, change in the form of the function, for the limiting 
values of x only ; and the indices i and 2 state that 
the actual limiting values are to be put in the place of 
the general expressions. The terms of the last line, 
finally, depend on the general change in the form of 
the function. Collecting all the terms, except those in 
the last line, under one designation a^ a , and calling 
the expression in parentheses in the last line /?, we 
have 

= or j a -(- C ' fi . 6y . dx. 

But this equation can be satisfied only if 

",-o = 0) 

and 

J l /3dydx = (2) 

For if each of the members were not equal to zero, 
each would be determined by the other. But the in- 
tegral of an indeterminate function cannot be expressed 
in terms of its limiting values only. Assuming, there- 
fore, that the equation 



J(56ydx = Q, 



The equa- holds generally good, its conditions can be satisfied, 

tion which . . , . ,. , 

solves the since oy is throughout arbitrary and its generality 01 
makes the form cannot be restricted, only by making /3 = 0. By 

function in 

question a the equation 

dP d~P d 3 P 

N- l - + 2 - tJLi 4. . . . = 0. . . (3), 

dx ^ dx* dx* T k J 

therefore, the form of the function y = q)(x) that makes 
the expression [fa. maximum or minimum is defined. 



maximum 
or mini- 



FORMAL DEVELOPMENT. 443 

Equation (3) was found by Euler. But Lagrange first 
showed the application of equation (1), for the deter- 
mination of a function by the conditions at its limits. 
By equation (3), which it must satisfy, the form of the 
function y=<p(x) is generally determined; but this 
equation contains a number of arbitrary constants, 
whose values are determined solely by the conditions 
at the limits. With respect to notation, Jellett rightly 
remarks, that the employment of the symbol d in the 
first two terms V x 6x % = V 6x Q of equation (1), (the 
form used by Lagrange,) is illogical, and he correctly 
puts for the increments of the independent variables 
the usual symbols dx lf dx Q . 

9. To illustrate the use to which these equations a practical 
may be put, let us seek the form of the function that of the use 

of these 
makes equations. 



All expressions except 



;st line. I 



-r \fx) 

a minimum the shortest line. Here 



P. = 



ay 

dV dx 



dx 



vanish in equation (3), and that equation becomes 
dP x jdx = ; which means that P x , and consequently 
its only variable, dy dx, is independent of x. Hence, 

ix = a, and y = a x -\- b, where a and b are con- 
stants. 

The constants a, b are determined by the values of 



444 THE SCIENCE OF MECHANICS. 

Develop- the limits. If the straight line passes through the 

mem of the 

illustration. points x , y and x v y x , then 

and as dx Q = dx x = 0, Sy = fiy x = 0, equation (i) 
vanishes. The coefficients S (dy/dx), d (d 2 y/dx 2 ), .... 
independently vanish. Hence, the values of a and b 
are determined by the equations (/) alone. 

If the limits x Q , x x only are given, buty Q ,)\ are 
indeterminate, we have ^/.r Q = dx x = 0, and equation 
(i) takes the form 

-^ (Jy, dr.0 = o, 

which, since Sy Q and dy x are arbitrary, can only be 
satisfied if a 0. The straight line is in this case 
y s= b, parallel to the axis of abscissae, and as b is inde- 
terminate, at any distance from it. 

It will be noticed, that equation (i) and the sub- 
sidiary conditions expressed in equation (m), with re- 
spect to the determination of the constants, generally 
complement each other. 

If 



*-/Wi+&) , <- 



\dxj 
i 

is to be made a minimum, the integration of the appro- 
priate form of (3) will give 



c 

y 



+ * 



2 

If Z is a minimum, then inZ also is a minimum, and 
the curve found will give, by rotation about the axis 
of abscissa?, the least surface of revolution. Further, 



FORMAL DEVELOPMEXT. 445 

to a minimum of Z the lowest position of the centre of 
gravity of a homogeneously heavy curve of this kind 
corresponds ; the curve is therefore a catenary. The 
determination of the constants c, c' is effected by means 
of the limiting conditions, as above. 

In the treatment of mechanical problems, a dis- variations 

. , and virtual 

tinction is made between the increments of coordinates dispiace- 

.... ments dis- 

that actually take place in time, namely, dx, dy, dz, anguished, 
and the possible displacements dx, Sy, 6z, considered, 
for instance, in the application of the principle of vir- 
tual velocities. The latter, as a rule, are not varia- 
tions ; that is, are not changes of value that spring 
from changes in the form of a function. Only when 
we consider a mechanical system that is a continuum, 
as for example a string, a flexible surface, an elastic 
body, or a liquid, are we at liberty to regard Sx, Sy, 
dz as indeterminate functions of the coordinates x, y, 
s, and are we concerned with variations. 

It is not our purpose in this work, to develop math- importance 

. . . of the cal- 

ematical theories, but simply to treat the purely phys- cuius of va- 

. r J r - v J riations for 

ical part of mechanics. But the history of the isoperi- mechanics, 
metrical problems and of the calculus of variations had 
to be touched upon, because these researches have ex- 
ercised a very considerable influence on the develop- 
ment of mechanics. Our sense of the general prop- 
erties of systems, and of properties of maxima and 
minima in particular, was much sharpened by these 
investigations, and properties of the kind referred to 
were subsequently discovered in mechanical systems 
with great facility. As a fact, physicists, since La- 
grange's time, usually express mechanical principles 
in a maximal or minimal form. This predilection 
would be unintelligible without a knowledge of the 
historical development. 



446 THE SCIENCE OE MECHANICS. 



THEOLOGICAL. ANIMISTIC, AND MYSTICAL POINTS OF VIEW 
IN MECHANICS. 

i. If, in entering a parlor in Germany, we happen 
to hear something said about some man being very 
pious, without having caught the name, we may fancy 
that Privy Counsellor X was spoken of, or Herr von 
Y ; we should hardly think of a scientific man of our 
acquaintance. It would, however, be a mistake to sup- 
pose that the want of cordiality, occasionally rising to 
embittered controversy, which has existed in our day 
between the scientific and the theological faculties, 
always separated them. A glance at the history of 
science suffices to prove the contrary. 
The con- People talk of the "conflict" of science and the- 

flictofsci- . 

ence and ology, or better of science and the church. It is in 
truth a prolific theme. On the one hand, we have the 
long catalogue of the sins of the church against pro- 
gress, on the other side a "noble army of martyrs," 
among them no less distinguished figures than Galileo 
and Giordano Bruno. It was only by good luck that 
Descartes, pious as he was, escaped the same fate. 
These things are the commonplaces of history; but it 
would be a great mistake to suppose that the phrase 
"warfare of science" is a correct description of its 
general historic attitude toward religion, that the only 
repression of intellectual development has come from 
priests, and that if their hands had been held off, grow-i 
ing science would have shot up with stupendous velo-j 
city. No doubt, external opposition did have to be 
fought ; and the battle with it was no child's play. 



FORMAL DEVELOPMENT 447 

Nor was any engine too base for the church to handle The stme- 

* gleofscien- 

in this struggle. She considered nothing but how to tistswuh 

30 , their own 

conquer ; and no temporal policy ever was conducted precon- 
so selfishly, so unscrupulously, or so cruelly. But in- ideas, 
vestigators have had another struggle on their hands, 
and by no means an easy one, the struggle with their 
own preconceived ideas, and especially with the notion 
that philosophy and science must be founded on the- 
ology. It was but slowly that this prejudice little by 
little was erased. 

2. But let the facts speak for themselves, while we Historical 

. examples. 

introduce the reader to a few historical personages. 

Napier, the inventor of logarithms, an austere Puri- 
tan, who lived in the sixteenth century, was, in addi- 
tion to his scientific avocations, a zealous theologian. 
Napier applied himself to some extremely curious 
speculations. He wrote an exegetical commentary on 
the Book of Revelation, with propositions and mathe- 
matical demonstrations. Proposition XXYL for ex- 
ample, maintains that the pope is the Antichrist ; propo- 
sition XXXVI declares that the locusts are the Turks 
and Mohammedans ; and so forth. 

Blaise Pascal (1623-1662), one of the most rounded 
geniuses to be found among mathematicians and phys- 
icists, was extremely orthodox and ascetical. So deep 
were the convictions of his heart, that despite the gen- 
tleness of his character, he once openly denounced at 
Rouen an instructor in philosophy as a heretic. The 
healing of his sister by contact with a relic most seri- 
ously impressed him, and he regarded her cure as a 
miracle. On these facts taken by themselves it might 
be wrong to lay great stress ; for his whole family were 
much inclined to religious fanaticism. But there are 
plenty of other instances of his religiosity. Such was 



44 8 



THE SCIENCE OF MECHANICS. 



P.lSC.ll. 



Otto von 
Guericke. 



his resolve, which was carried out, too, to abandon 
altogether the pursuits of science and to devote his life 
solely to the cause of Christianity. Consolation, he 
used to say, he could find nowhere but in the teachings 
of Christianity ; and all the wisdom of the world availed 
him not a whit. The sincerity of his desire for the 
conversion of heretics is shown in his Lettres provin- 
lialcs, where he vigorously declaims against the dread 
ful subtleties that the doctors of the Sorbonne had 
devised, expressly to persecute the Jansenists. Very 
remarkable is Pascal's correspondence with the theo- 
logians of his time ; and a modern reader is not a little 
surprised at finding this great "scientist" seriously 
discussing in one of his letters whether or not the Devil 
was able to work miracles. 

Otto von Guericke, the inventor of the air-pump, 
occupies himself, at the beginning of his book, now 
little over two hundred years old, with the miracle of 
Joshua, which he seeks to harmonise with the ideas 
of Copernicus. In like manner, we find his researches 
on the vacuum and the nature of the atmosphere in- 
troduced by disquisitions concerning the location of 
heaven, the location of hell, and so forth. Although 
Guericke really strives to answer these questions as ra- 
tionally as he can, still we notice that they give him 
considerable trouble, questions, be it remembered, 
that to-day the theologians themselves would consider 
absurd. Yet Guericke was a man who lived after the 
Reformation ! 

The giant mind of Newton did not disdain to employ 
itself on the interpretation of the Apocalypse. On such 
subjects it was difficult for a sceptic to converse with 
him. When Halley once indulged in a jest concerning 
theological questions, he is said to have curtly repulsed 



FORMAL DEVELOPMENT. 449 

him with the remark : "I have studied these things : Xewtonand 

Leibnitz. 

you have not : 

Wc need not tarry by Leibniz, the inventor of the 
best of all possible worlds and of pre-established har- 
mony inventions which Voltaire disposed of in Can- 
dide, a humorous novel with a deeply philosophical pur- 
pose. But everybody knows that Leibnyrz was almost 
if not quite as much a theologian, as a man of science. 

Let us turn, however, to the last century. Euler, in Euler. 
his Letters to a German Princess, deals with theologico- 
philosophical problems in the midst of scientific ques- 
tions. He speaks of the difficulty involved in explaining 
the interaction of body and mind, due to the total 
diversity of these two phenomena, a diversity to his 
mind undoubted. The system of occasionalism, devel- 
oped by Descartes and his followers, agreeably to which 
God executes for every purpose of the soul, (the soul it- 
self not being able to do so,) a corresponding movement 
of the body, does not quite satisfy him. He derides, 
also, and not without humor, the doctrine of pre- 
established harmony, according to which perfect agree- 
ment was established from the beginning between the 
movements of the body and the volitions of the soul, 
although neither is in any way connected with the 
other, just as there is harmony between two different 
but like-constructed clocks. He remarks, that in this 
view his own body is as foreign to him as that of a 
rhinoceros in the midst of Africa, which might just as 
well be in pre-established harmony with his soul as 
its own. Let us hear his own words. In his day, Latin 
was almost universally written. When a German 
scholar wished to be especially condescending, he 
wrote in French : " Si dans le cas d'un dereglement 
''de mon corps Dieu ajustait celui d'un rhinoceros, 



450 THE SCIENCE OF MECHANICS. 

"en sorte que ses mouvements fussent tellement d'ac- 
" cord avec les ordres de mon ame, qu'il levat la patte 
"au moment que je voudrais lever la main, et ainsi 
" des autres operations, ce serait alors mon corps. Je 
" me trouverais subitement dans la forme d'un rhino- 
ceros au milieu de l'Afrique, mais non obstant cela 
"mon ame continuerait les meme operations. J'aurais 
" galement l'honneur d'^crire a V. A., mais je ne sais 
"pas comment elle recevrait mes lettres." 
Euier's One would almost imagine that Euler, here, had been 

theological . 

proclivities tempted to play Voltaire. And yet, apposite as was 
his criticism in this vital point, the mutual action of 
body and soul remained a miracle to him, still. But he 
extricates himself, however, from the question of the 
freedom of the will, very sophistically. To give some 
idea of the kind of questions which a scientist was per- 
mitted to treat in those days, it may be remarked that 
Euler institutes in his physical "Letters" investiga- 
tions concerning the nature of spirits, the connection 
between body and soul, the freedom of the will, the 
influence of that freedom on physical occurrences, 
prayer, physical and moral evils, the conversion of sin- 
ners, and such like topics ; and this in a treatise full 
of clear physical ideas and not devoid of philosophical 
ones, where the well-known circle-diagrams of logic 
have their birth-place. 

character 3. Let these examples of religious physicists suffice. 

logical We have selected them intentionally from among the 

the great in- foremost of scientific discoverers. The theological pro- 
clivities which these men followed, belong wholly to 
their innermost private life. They tell us openly things 
which they are not compelled to tell us, things about 
which they might have remained silent. What they 
utter are not opinions forced upon them from without ; 



FORMAL DEVELOPMEXT. 451 

they are their own sincere views. They were not con- 
scious of any theological constraint. In a court which 
harbored a Lamettrie and a Voltaire, Eulerhad no rea- 
son to conceal his real convictions. 

According to the modern notion, these men should character 

1 11 1 j- oftheirage. 

at least have seen that the questions they discussed 
did not belong under the heads where they put them, 
that they were not questions of science. Still, odd as 
this contradiction between inherited theological beliefs 
and independently created scientific convictions seems 
to us, it is no reason for a diminished admiration of 
those leaders of scientific thought. Nay, this very fact 
is a proof of their stupendous mental power : they were 
able, in spite of the contracted horizon of their age, to 
which even their own aperrus were chiefly limited, to 
point out the path to an elevation, where our genera- 
tion has attained a freer point of view. 

Every unbiassed mind must admit that the age in 
which the chief development of the science of mechan- 
ics took place, was an age of predominantly theological 
cast. Theological questions were excited by everything, 
and modified everything. No wonder, then, that me- 
chanics took the contagion. But the thoroughness with 
which theological thought thus permeated scientific 
inquiry, will best be seen by an examination of details. 

4. The impulse imparted in antiquity to this direc- Galileo's 

A " J researches 

tion of thought bv Hero and Pappus has been alluded on the 

J rr strength of 

to in the preceding chapter. At the beginning of the materials, 
seventeenth century we find Galileo occupied with prob- 
lems concerning the strength of materials. He shows 
that hollow tubes offer a greater resistance to flexure 
than solid rods of the same length and the same quantity 
of material, and at once applies this discovery to the 
explanation of the forms of the bones of animals, which 



452 THE SCIENCE OF MECHANICS. 

are usually hollow and cylindrical in shape. The phe- 
nomenon is easily illustrated by the comparison of a 
flatly folded and a rolled sheet of paper. A horizontal 
beam fastened at one extremity and loaded at the other 
may be remodelled so as to be thinner at the loaded 
end without any loss of stiffness and with a consider- 
able saving of material. Galileo determined the form of 
a beam of equal resistance at each cross-section. He 
also remarked that animals of similar geometrical con- 
struction but of considerable difference of size would 
comply in very unequal proportions with the laws of 
resistance. 
Evidences The forms of bones, feathers, stalks, and other or- 

of design . ... 

in nature, game structures, adapted, as they are, m their minut- 
est details to the purposes they serve, are highly cal- 
culated to make a profound impression on the thinking 
beholder, and this fact has again and again been ad- 
duced in proof of a supreme wisdom ruling in nature. 
Let us examine, for instance, the pinion-feather of a 
bird. The quill is a hollow tube diminishing in thick- 
ness as we go towards the end, that is, is a body of 
equal resistance. Each little blade of the vane re- 
peats in miniature the same construction. It would \ 
require considerable technical knowledge even to imi- 
tate a feather of this kind, let alone invent it. We : 
should not forget, however, that investigation, and i 
not mere admiration, is the office of science. We 
know how Darwin sought to solve these problems, by 
the theory of natural selection. That Darwin's solution 
is a complete one, may fairly be doubted ; Darwin him- 
self questioned it. All external conditions would be 
powerless if something were not present that admitted I 
of variation. But there can be no question that his ' 
theory is the first serious attempt to replace mere ad- 



FORMAL DEVELOPMENT. 453 

miration of the adaptations of organic nature by seri- 
ous inquiry into the mode of their origin. 

Pappus's ideas concerning the cells of honeycombs The ceils r.f 

.... , the honey- 

were the subject of animated discussion as late as the comb. 

eighteenth century. In a treatise, published in 1865, 
entitled Homes Without Hands (p. 428), Wood substan- 
tially relates the following : " Maraldi had been struck 
with the great regularity of the cells of the honey- 
comb. He measured the angles of the lozenge-shaped 
plates, or rhombs, that form the terminal walls of the 
cells, and found them to be respectively io928' and 
70 32'. Reaumur, convinced that these angles were in 
some way connected with the economy of the cells, 
requested the mathematician Konig to calculate the 
form of a hexagonal prism terminated by a pyramid 
composed of three equal and similar rhombs, which 
would give the greatest amount of space with a given 
amount of material. The answer was, that the angles 
should be io926' and 7o34'. The difference, accord- 
ingly, was two minutes. Maclaurin,* dissatisfied with 
this agreement,repeated Maraldi's measurements, found 
them correct, and discovered, in going over the calcu- 
lation, an error in the logarithmic table employed by 
Konig. Not the bees, but the mathematicians were 
wrong, and the bees had helped to detect the error ! " 
Any one who is acquainted with the methods of meas- 
uring crystals and has seen the cell of a honeycomb, 
with its rough and non-reflective surfaces, will question 
whether the measurement of such cells can be executed 
with a probable error of only two minutes. f So, we 
must take this story as a sort of pious mathematical 

* Philosophical Transactions for 1743. Trans. 

t But see G. F. Maraldi in the Mimoires de facadimie for 1712. It is, how- 
ever, now well known the cells vary considerably. See Chauncey Wright, 
Philosophi-al Discussions, 1877, p. 311. Trans. 



454 THE SCIENCE OF MECHANICS. 

fairy-tale, quite apart from the consideration that noth- 
ing would follow from it even were it true. Besides, 
from a mathematical point of view, the problem is too 
imperfectly formulated to enable us to decide the ex- 
tent to which the bees have solved it. 
other The ideas of Hero and Fermat, referred to in the 

instances. . 

previous chapter, concerning the motion of light, at 
once received from the hands of Leibniz a theolog- 
ical coloring, and played, as has been before mentioned, 
a predominant role in the development of the calculus 
of variations. In Leibniz's correspondence with John 
Bernoulli, theological questions are repeatedly dis- 
cussed in the very midst of mathematical disquisitions. 
Their language is not unfrequently couched in biblical 
pictures. Leibniz, for example, says that the problem 
of the brachistochrone lured him as the apple had lured 
Eve. 
Thetheo- Maupertuis, the famous president of the Berlin 

K > iii c 1 1 1 K c r~ 

neiofthe Academy, and a friend of Frederick the Great, gave 

principle of . . . . 

least ac- a new impulse to the theologising bent of physics by 
the enunciation of his principle of least action. In the 
treatise which formulated this obscure principle, and 
which betrayed in Maupertuis a woeful lack of mathe- 
matical accuracy, the author declared his principle to be 
the one which best accorded with the wisdom of the 
Creator. Maupertuis was an ingenious man, but not a 
man of strong, practical sense. This is evidenced by 
the schemes he was incessantly devising : his bold prop- 
ositions to found a city in which only Latin should be 
spoken, to dig a deep hole in the earth to find new 
substances, to institute psychological investigations by 
means of opium and by the dissection of monkeys, to 
explain the formation of the embryo by gravitation, and 
so forth. He was sharply satirised by Voltaire in the 



FORMAL DEVELOPMEXT. 455 

Histoire du docteur Akakia, a work which led, as we 
know, to the rupture between Frederick and Voltaire. 

Maupertuis's principle would in all probability soon Euiersre- 

r r * tention of 

have been forgotten, had Euler not taken up the sug- the theoiog- 

. , .... ical basis of 

gestion. Euler magnanimously left the principle its this prin- 

r j ciple. 

name. Maupertuis the glory of the invention, and con- 
verted it into something new and really serviceable. 
What Maupertuis meant to convey is very difficult to 
ascertain. What Euler meant may be easily shown by 
simple examples. If a body is constrained to move on a 
rigid surface, for instance, on the surface of the earth, it 
will describe when an impulse is imparted to it, the 
shortest path between its initial and terminal positions. 
Any other path that might be prescribed it, would be 
longer or would require a greater time. This principle 
finds an application in the theory of atmospheric and 
oceanic currents. The theological point of view, Euler 
retained. He claims it is possible to explain phenomena, 
not only from their physical causes, but also from their 
purposes. "As the construction of the universe is the 
"most perfect possible, being the handiwork of an 
"all-wise Maker, nothing can be met with in the world 
"in which some maximal or minimal property is not 
"displayed. There is, consequently, no doubt but 
"that all the effects of the world can be derived by 
"the method of maxima and minima from their final 
"causes as well as from their efficienc ones."* 

5. Similarly, the notions of the constancy of the 
quantity of matter, of the constancy of the quantity of 

* " Quum enim mundi universi fabrica sit perfectissima, atque a creatore 
sapientissimo absoluta. nihil omnino in mundo contingit, in quo non maximi 
minimive ratio quaepiain eluceat: quam ob rem dubium prorsus est nullum, 
quin omnes mundi effectus ex causis finalibus. ope methodi maximorum et 
minimorum, aeque feliciter determinari quaeant. atque ex ipsis causis efficien- 
tibus." {Methodus inveniendi lineas curvas maximi minimive proprietate 
gaudcntes. Lausanne, 1744.) 



456 THE SCIENCE OF MECHANICS. 

The central motion, of the indestructibility of work or energy, con- 
notions Of . . . , ill- J 1 

modern ceptions which completely dominate modern physics, 
mainly of all arose under the influence of theological ideas. The 

theological . . .,,,.... - 

origin. notions in question had their origin in an utterance ot 
Descartes, before mentioned, in the Principles of Philos- 
opiiy, agreeably to which the quantity of matter and mo- 
tion originally created in the world, such being the 
only course compatible with the constancy of the Crea- 
tor, is always preserved unchanged. The conception 
of the manner in which this quantity of motion should 
be calculated was very considerably modified in the 
progress of the idea from Descartes to Leibnitz, and to 
their successors, and as the outcome of these modifi- 
cations the doctrine gradually and slowly arose which 
is now called the " law of the conservation of energy." 
But the theological background of these ideas only 
slowly vanished. In fact, at the present day, we still 
meet with scientists who indulge in self-created mys- 
ticisms concerning this law. 
Gradual During the entire sixteenth and seventeenth centu- 

from the ries, down to the close of the eighteenth, the prevail- 

theological . .... - . . /- j n i - 1 

point of ing inclination of inquirers was, to find in all physical 
laws some particular disposition of the Creator. But 
a gradual transformation of these views must strike 
the attentive observer. Whereas with Descartes and 
Leibnitz physics and theology were still greatly inter- 
mingled, in the subsequent period a distinct endeavor 
is noticeable, not indeed wholly to discard theology, 
yet to separate it from purely physical questions. Theo- 
logical disquisitions were put at the beginning or rele- 
gated to the end of physical treatises. Theological 
speculations were restricted, as much as possible, to 
the question of creation, that, from this point onward, 
the way might be cleared for physics. 



view. 



FORMAL DEVELOPMEXT. 457 

Towards the close of the eighteenth century a re- ultimate 
markable change took place, a change which was emancipa- 
apparently an abrupt departure from the current trend physics 
of thought, but in reality was the logical outcome of o gy . 
the development indicated. After an attempt in a 
youthful work to found mechanics on Euler's principle 
of least action, Lagrange, in a subsequent treatment 
of the subject, declared his intention of utterly disre- 
garding theological and metaphysical speculations, as 
in their nature precarious and foreign to science. He 
erected a new mechanical system on entirely different 
foundations, and no one conversant with the subject 
will dispute its excellencies. All subsequent scientists 
of eminence accepted Lagrange's view, and the pres- 
ent attitude of physics to theology was thus substan- 
tially determined. 

6. The idea that theology and physics are two dis- The mod- 

J * J em ideal 

tinct branches of knowledge, thus took, from its first always the 

\ attitude of 

germination in Copernicus till its final promulgation th e greatest 

. r . c inquirers. 

by Lagrange, almost two centuries to attain clearness 
in the minds of investigators. At the same time it 
cannot be denied that this truth was always clear to 
the greatest minds, like Newton. Newton never, de- 
spite his profound religiosity, mingled theology with 
the questions of science. True, even he concludes his 
Optics, whilst on its last pages his clear and luminous 
intellect still shines, with an exclamation of humble 
contrition at the vanity of all earthly things. But his 
optical researches proper, in contrast to those of Leib- 
nitz, contain not a trace of theology. The same may 
be said of Galileo and Huygens. Their writings con- 
form almost absolutely to the point of view of La- 
grange, and may be accepted in this respect as class- 
ical. But the general views and tendencies of an age 



458 THE SCIENCE OF MECHANICS. 

must not be judged by its greatest, but by its average, 

minds. 
The theo- To comprehend the process here portrayed, the gen- 

ceptionof eral condition of affairs in these times must be consid- 

thc world . 

natural and ered. It stands to reason that in a stage of civilisation 

explain- . . . 

able. in which religion is almost the sole education, and the 

only theory of the world, people would naturally look 
at things in a theological point of view, and that they 
would believe that this view was possessed of compe- 
tency in all fields of research. If we transport ourselves 
back to the time when people played the organ with 
their fists, when they had to have the multiplication table 
visibly before them to calculate, when they did so much 
with their hands that people now-a-days do with their 
heads, we shall not demand of such a time that it 
should critically put to the test its own views and the- 
ories. With the widening of the intellectual horizon 
through the great geographical, technical, and scien- 
tific discoveries and inventions of the fifteenth and six- 
teenth centuries, with the opening up of provinces in 
which it was impossible to make any progress with the 
old conception of things, simply because it had been 
formed prior to the knowledge of these provinces, this 
bias of the mind gradually and slowly vanished. The 
great freedom of thought which appears in isolated 
cases in the early middle ages, first in poets and then 
in scientists, will always be hard to understand. The en- 
lightenment of those days must have been the work of a 
few very extraordinary minds, and can have been bound 
to the views of the people at large by but very slender 
threads, more fitted to disturb those views than to re- 
form them. Rationalism does not seem to have gained 
a broad theatre of action till the literature of the eigh- 
teenth century. Humanistic, philosophical, historical, 



FORMAL DEVELOPME.\ T. 459 

and physical science here met and gave each other 
mutual encouragement. All who have experienced, in 
part, in its literature, this wonderful emancipation of 
the human intellect, will feel during their whole lives a 
deep, elegiacal regret for the eighteenth century. 

7. The old point of view, then, is abandoned. Its The en- 

. . , , lighten- 

history is now detectible only in the form of the me- mem of the 

J . new views. 

chanical principles. And this form will remain strange 
to us as long as we neglect its origin. The theological 
conception of things gradually gave way to a more 
rigid conception ; and this was accompanied with a 
considerable gain in enlightenment, as we shall now 
briefly indicate. 

When we say light travels by the paths of shortest 
time, we grasp by such an expression many things. 
But we do not know as yet why light prefers paths of 
shortest time. We forego all further knowledge of the 
phenomenon, if we find the reason in the Creator's wis- 
dom. We of to-day know, that light travels by all 
paths, but that only on the paths of shortest time do 
the waves of light so intensify each other that a per- 
ceptible result is produced. Light, accordingly, only 
appears to travel by the paths of shortest time. After Extrava- 

. . gance as 

the prejudice which prevailed on these questions had well as 

x J * . . economy in 

been removed, cases were immediately discovered in nature, 
which by the side of the supposed economy of nature 
the most striking extravagance was displayed. Cases 
of this kind have, for example, been pointed out by 
Jacobi in connection with Euler's principle of least ac- 
tion. A great many natural phenomena accordingly 
produce the impression of economy, simply because 
they visibly appear only when by accident an econom- 
ical accumulation of effects take place. This is the 
same idea in the province of inorganic nature that Dar- 



460 



THE SCIENCE OF MECHANICS. 



Explana- 
tion of max 

im.-il and 
minimal 
effects. 



Points of 
identity in 
the theolog- 
ical and 
scientific 
concep- 
tions. 



win worked out in the domain of organic nature. We 
facilitate instinctively our comprehension of nature by 
applying to it the economical ideas with which we are 
familiar. 

Often the phenomena of nature exhibit maximal 
or minimal properties because when these greatest or 
least properties have been established the causes of all 
further alteration are removed. The catenary gives 
the lowest point of the centre of gravity for the simple 
reason that when that point has been reached all fur- 
ther descent of the system's parts is impossible. Li- 
quids exclusively subjected to the action of molecular 
forces exhibit a minimum of superficial area, because 
stable equilibrium can only subsist when the molecular 
forces are able to effect no further diminution of super- 
ficial area. The important thing, therefore, is not the 
maximum or minimum, but the removal of work ; work- 
being the factor determinative of the alteration. It 
sounds much less imposing but is much more elucida- 
tory, much more correct and comprehensive, instead 
of speaking of the economical tendencies of nature, to 
say : "So much and so much only occurs as in virtue 
of the forces and circumstances involved can occur." 

The question may now justly be asked, If the point 
of view of theology which led to the enunciation of the 
principles of mechanics was utterly wrong, how comes 
it that the principles themselves are in all substantial 
points correct ? The answer is easy. In the first place, 
the theological view did not supply the contents of the 
principles, but simply determined their guise; their mat- 
ter was derived from experience. A similar influence 
would have been exercised by any other dominant type 
of thought, by a commercial attitude, for instance, such 
as presumably had its effect on Stevinus's thinking. In 



FORMAL DEVELOPMEXT. 461 

the second place, the theological conception of nature 
itself owes its origin to an endeavor to obtain a more 
comprehensive view of the world ; the very same en- 
deavor that is at the bottom of physical science. Hence, 
even admitting that the physical philosophy of theology 
is a fruitless achievement, a reversion to a lower state of 
scientific culture, we still need not repudiate the sound 
root from which it has sprung and which is not differ- 
ent from that of true physical inquiry. 

In fact, science can accomplish nothing by the con- Necessity 
' v . . of a con- 

sideration of individual 'facts ; from time to time it must stam con- 

sideration 

cast its glance at the world as a whole. Galileo's of the ah. 

, in research 

laws of falling bodies, Huygens's principle of vis viva, 
the principle of virtual velocities, nay, even the con- 
cept of mass, could not, as we saw, be obtained, ex- 
cept by the alternate consideration of individual facts 
and of nature as a totality. We may, in our men- 
tal reconstruction of mechanical processes, start from 
the properties of isolated masses (from the elementary 
or differential laws), and so compose our pictures of 
the processes ; or, we may hold fast to the properties 
of the system as a whole (abide by the integral laws). 
Since, however, the properties of one mass always in- 
clude relations to other masses, (for instance, in ve- 
locity and acceleration a relation of time is involved, 
that is, a connection with the whole world,) it is mani- 
fest that purely differential, or elementary, laws do not 
exist. It would be illogical, accordingly, to exclude 
as less certain this necessary view of the All, or of the 
more general properties of nature, from our studies. 
The more general a new principle is and the wider its 
scope, the more perfect tests will, in view of the possi- 
bility of error, be demanded of it. 

The conception of a will and intelligence active in 



462 THE SCIENCE OF MECHANICS. 

Pagan ideas nature is by no means the exclusive property of Chris- 

and prac- . . .... . 

tices rife in tian monotheism. On the contrary, this idea is a quite 

the modern ... . ..... ._ 

world. iamihar one to paganism and fetishism. Paganism, 
however, finds this will and intelligence entirely in in- 
dividual phenomena, while monotheism seeks it in the 
All. Moreover, a pure monotheism does not exist. 
The Jewish monotheism of the Bible is by no means 
free from belief in demons, sorcerers, and witches ; 
and the Christian monotheism of mediaeval times is 
even richer in these pagan conceptions. We shall not 
speak of the brutal amusement in which church and 
state indulged in the torture and burning of witches, 
and which was undoubtedly provoked, in the majority 
of cases, not by avarice but by the prevalence of the 
ideas mentioned. In his instructive work on Primitive 
Culture Tylor has studied the sorcery, superstitions, 
and miracle-belief of savage peoples, and compared 
them with the opinions current in mediaeval times con- 
cerning witchcraft. The similarity is indeed striking. 
The burning of witches, which was so frequent in 
Europe in the sixteenth and seventeenth centuries, is 
to-day vigorously conducted in Central Africa. Even 
now and in civilised countries and among cultivated 
people traces of these conditions, as Tylor shows, still 
exist in a multitude of usages, the sense of which, with 
our altered point of view, has been forever lost. 

8. Physical science rid itself only very slowly of 
these conceptions. The celebrated work of Giambatista 
della Porta, Magia naturalis, which appeared in 1558, 
though it announces important physical discoveries, is 
yet filled with stuff about magic practices and demono- 
logical arts of all kinds little better than those of a red- 
skin medicine-man. Not till the appearance of Gil- 
bert's work, De magnete (in 1600), was any kind of re- 



FORMAL DEVELOPMEXT. 463 

striction placed on this tendency of thought. When we Animistic 
reflect that even Luther is said to have had personal science, 
encounters with the Devil, that Kepler, whose aunt had 
been burned as a witch and whose mother came near 
meeting the same fate, said that witchcraft could not 
be denied, and dreaded to express his real opinion of 
astrology, we can vividly picture to ourselves the 
thought of less enlightened minds of those ages. 

Modern physical science also shows traces of fetish- 
ism, as Tylor well remarks, in its "forces." And the 
hobgoblin practices of modern spiritualism are ample 
evidence that the conceptions of paganism have not 
been overcome even by the cultured society of to-day. 

It is natural that these ideas so obstinately assert 
themselves. Of the many impulses that rule man 
with demoniacal power, that nourish, preserve, and 
propagate him, without his knowledge or supervision, 
of these impulses of which the middle ages present 
such great pathological excesses, only the smallest 
part is accessible to scientific analysis and conceptual 
knowledge. The fundamental character of all these 
instincts is the feeling of our oneness and sameness 
with nature ; a feeling that at times can be silenced 
but never eradicated by absorbing intellectual occupa- 
tions, and which certainly has a sound basis, no matter 
to what religious absurdities it may have given rise. 

9. The French encyclopaedists of the eighteenth 
century imagined they were not far from a final ex- 
planation of the world by physical and mechanical prin- 
ciples ; Laplace even conceived a mind competent to 
foretell the progress of nature for all eternity, if but the 
masses, their positions, and initial velocities were given. 
In the eighteenth century, this joyful overestimation of 
the scope of the new physico-mechanical ideas is par- 



464 THE SCIENCE OE MECHANICS. 

Ovc-esti- donable. Indeed, it is a refreshing, noble, and ele- 

In.ition of 

theme- vating spectacle ; and we can deeply sympathise with 
chanical .. . ..... 

view". this expression of intellectual joy, so unique m history. 

But now, after a century has elapsed, after our judg- 
ment has grown more sober, the world-conception of the 
encyclopaedists appears to us as a mechanical mythology 
in contrast to the animistic of the old religions. Both 
views contain undue and fantastical exaggerations of 
an incomplete perception. Careful physical research 
will lead, however, to an analysis of our sensations. 
We shall then discover that our hunger is not so essen- 
tially different from the tendency of sulphuric acid for 
zinc, and our will not so greatly different from the 
pressure of a stone, as now appears. We shall again 
feel ourselves nearer nature, without its being neces- 1 
sary that we should resolve ourselves into a nebulous 
and mystical mass of molecules, or make nature a 
haunt of hobgoblins. The direction in which this en- j 
lightenment is to be looked for, as the result of long) 
and painstaking research, can of course only be sur- 
mised. To anticipate the result, or even to attempt to, 
introduce it into any scientific investigation of to-day, I 
would be mythology, not science. 
Pretensions Physical science does not pretend to be a completi 
tude of view of the world ; it simply claims that it is work 
science. toward such a complete view in the future. The high 1 
est philosophy of the scientific investigator is precisel; 
this toleration of an incomplete conception of the worl< 
and the preference for it rather than an apparently pet 
feet, but inadequate conception. Our religious opir. 
ions are always our own private affair, as long as we d 
not obtrude them upon others and do not apply then 
to things which come under the jurisdiction of a diffe 
ent tribunal. Physical inquirers themselves enterta; 



FORMAL DEVELOPMEXT. 465 

the most diverse opinions on this subject, according to 
the range of their intellects and their estimation of the 
consequences. 

Physical science makes no investigation at all into 
things that are absolutely inaccessible to exact investi- 
gation, or as yet inaccessible to it. But should prov- 
inces ever be thrown open to exact research which are 
now closed to it, no well-organised man, no one who 
cherishes honest intentions towards himself and others, 
will any longer then hesitate to countenance inquiry 
with a view to exchanging his opinion regarding such 
provinces for positive knowledge of them. 

When, to-day, we see society waver, see it change Results of 

. . . . ... the incom- 

its views on the same question according to its mood and pietenessof 

r 1 ii-ii r our view cf 

tne events of the week, like the register of an organ, when the world, 
we behold the profound mental anguish which is thus 
produced, we should know that this is the natural and 
necessary outcome of the incompleteness and transi- 
tional character of our philosophy. A competent view 
of the world can never be got as a gift ; we must ac- 
quire it by hard work. And only by granting free sway 
to reason and experience in the provinces in which they 
alone are determinative, shall we, to the weal of man- 
kind, approach, slowly, gradually, but surely, to that 
ideal of a monistic view of the world which is alone 
compatible with the economy of a sound mind. 

in. 

ANALYTICAL MECHANICS. 

1 . The mechanics of Newton are purely geometrical. The geo- 

T-T A A i--i r . . . metrical 

ne aeauces his theorems from his initial assumptions mechanics 

. , , of Newton. 

entirely by means of geometrical constructions. His 
procedure is frequently so artificial that, as Laplace 



466 THE SCIENCE OF MECHANICS. 

remarked, it is unlikely that the propositions were dis- 
covered in that way. We notice, moreover, that the 
expositions of Newton are not as candid as those of 
Galileo and Huygens. Newton's is the so-called syn- 
thetic method of the ancient geometers. 
Analytic When we deduce results from given suppositions, 

mechanics. . 

the procedure is called synthetic. When we seek the 
conditions of a proposition or of the properties of a fig- 
ure, the procedure is analytic. The practice of the latter 
method became usual largely in consequence of the 
application of algebra to geometry. It has become 
customary, therefore, to call the algebraical method 
generally, the analytical. The term " analytical me- 
chanics," which is contrasted with the synthetical, or 
geometrical, mechanics of Newton, is the exact equiva- 
lent of the phrase "algebraical mechanics." 
Euler and 2. The foundations of analytical mechanics were 

Maclau- . _ 

rinscon- laid by Euler {Mechanica, sive Motus Scicntia Analytice 

tributions. _, __^ , . 

Exposita, St. Petersburg, 1736). But while Euler s 
method, in its resolution of curvilinear forces into tan- 
gential and normal components, still bears a trace of 
the old geometrical modes, the procedure of Maclaurin 
(A Complete System of Fluxions, Edinburgh, 1 742) marks 
a very important advance. This author resolves all 
forces in three fixed directions, and thus invests the 
computations of this subject with a high degree of 
symmetry and perspicuity. 
Lagrange's 3. Analytical mechanics, however, was brought to 

perfection .... . ...,_ T 

of the its highest degree of perfection by Lagrange. La- 
grange's aim is (Mecanique analytique, Paris, 1788) to 
dispose once for alloi the reasoning necessary to resolve 
mechanical problems, by embodying as much as pos- 
sible 'of it in a single formula. This he did. Every case 
that presents itself can now be dealt with by a very 



science. 



FORMAL DEVELOPMENT. 467 

simple, highly symmetrical and perspicuous schema ; 
and whatever reasoning is left is performed by purely 
mechanical methods. The mechanics of Lagrange 
is a stupendous contribution to the economy of 
thought. 

In statics, Lagrange starts from the principle ofstaucs 

.... _ , . ... founded on 

virtual velocities. On a number of material points the pnnci- 
m x , m 2 , m 3 . . . ., definitely connected with one another, tuai veioci- 
are impressed the forces P x , P 2 , P 3 . . . . If these 
points receive any infinitely small displacements p xt 
p 2 , / 3 . . . . compatible with the connections of the sys- 
tem, then for equilibrium 2 Pp = ; where the well- 
known exception in which the equality passes into an 
inequality is left out of account. 

Now refer the whole system to a set of rectangular 
coordinates. Let the coordinates of the material points 
be x v y x , z x , x 2 , v 2 , z 2 .... Resolve the forces into 
the components X x , Y x , Z x , X a , Y 2 , Z 2 . . . . parallel 
to the axes of coordinates, and the displacements into 
the displacements dx lf dv x , 6s x , 6x 2 , 6v 2 , 6z 2 . . . ., 
also parallel to the axes. In the determination of the 
work done only the displacements of the point of appli- 
cation in the direction of each force-component need 
be considered for that component, and the expression 
of the principle accordingly is 

2(X6x+ Y6y+Z6s) = (1) 

where the appropriate indices are to be inserted for 
the points, and the final expressions summed. 

The fundamental formula of dvnamics is derived Dynamics 

r y^. . . on the prin- 

irom D Alembert's principle. On the material points cipie of 
m i> i 2 , m a . . . ., having the coordinates x x ,y x , z x , x 2 , ben. 

_, . . . . the force-components X x , Y x , Z x , X.,, Y 2 , 
Z 2 . . . . act. But, owing to the connections of the 



468 THE SCIENCE OF MECHANICS. 

system's parts, the masses undergo accelerations, which 
are those of the forces. 

d 2 x. d 2 v, d 2 z. 

These are called the effective forces. But the impressed 
forces, that is, the forces which exist by virtue of the 
laws of physics, X, Y, Z . . . . and the negative of these 
effective forces are, owing to the connections of the 
system, in equilibrium. Applying, accordingly, the 
principle of virtual velocities, we get 

z -'\M= (2 > 

Discussion 4. Thus, Lagrange conforms to tradition in making 
grangers statics precede dynamics. He was by no means com- 
pelled to do so. On the contrary, he might, with equal 
propriety, have started from the proposition that the 
connections, neglecting their straining, perform no 
work, or that all the possible work of the system is due 
to the impressed forces. In the latter case he would 
have begun with equation (2), which expresses this 
fact, and which, for equilibrium (or non-accelerated 
motion) reduces itself to (1) as a particular case. This 
would have made analytical mechanics, as a system, 
even more logical. 

Equation (1), which for the case of equilibrium 
makes the element of the work corresponding to the 
assumed displacement = 0, gives readily the results 
discussed in page 69. If 

ax dy dz 



method. 



FORMAL DEVELOPMEXT. 469 

that is to say. if A". V, Z are the partial differential co- 
efficients of one and the same function of the coordi- 
nates of position, the whole expression under the sign 
of summation is the total variation, 6 V, of V. If the 
latter is = 0, I ' is in general a maximum or a minimum. 

s. We will now illustrate the use of equation (1) by indication 

..." of the Ren- 

a simple example. If all the points of application of the erai steps 

r r * r , for the solo- 

forces are independent of each other, no problem is tion of stat- 

. . ... ,ca ' prob- 

presented. Each point is then in equilibrium onlyienis. 
when the forces impressed on it, and consequently 
their components, are = 0. All the displacements 6\v. 
. . . are then wholly arbitrary, and equation 
an subsist only provided the coefficients of all the 
displacements Sx, 6\\ Sz. . . . are equal to zero. 

But if equations obtain between the coordinates of 
the several points, that is to say, if the points are sub- 
ject to mutual constraints, the equations so obtaining 
will be of the form F(x lt y x . ,, z 2 . . . .) = 0, 

or. more briefly, of the form J?= 0. Then equations 
also obtain between the displacements, of the form 

dF dF dF JF 

^ x * x ^dT x Sy ^dT x Sz ^dxi 8x * + '=*' 

which we shall briefly designate asZ^^^O. If the 
m consist of n points, we shall have 3 coordi- 
nates, and equation (1) will contain 3// magnitudes 
Sx. dr. 6: . . . . If, further, between the coordinates 
m equations of the form F= subsist, then m equa- 
tions of the form DF= will be simultaneously given 
between the variations 6\v, Sv, Sz. . . . By these 
equations /// variations can be expressed in terms of the 
remainder, and so inserted in equation (1). In (1), 
therefore, there are left 3 n /// arbitrary displace- 
ments,, whose coefficients are put = 0. There are thus 



47Q 



THE SCIENCE OF MECHANICS. 



A statical 
example. 




Obtained between the forces and the coordinates 3 ;/ m 
equations, to which the /// equations {F= 0) must be 
added. We have, accordingly, in all, -$n equations, 
which are sufficient to determine the 3 coordinates of 
the position of equilibrium, provided the forces are 
given and only the form of the system's equilibrium is 
sought. 

But if the form of the system is given and the forces 
are sought that maintain equilibrium, the question is 
indeterminate. We have then, to determine 3 n force- 
components, only 3 m equa- 
tions; the ;// equations (^=0) 
not containing the force-compo- 
nents. 

As an example of this man- 
ner of treatment we shall select 
a lever OM=a, free to rotate 
about the origin of coordinates 
in the plane XY, and having at its end a second, simi- 
lar lever MN = b. At M and N, the coordinates of 
which we shall call x, v and x x , y x , the forces X, Kand 
X xt Y x are applied. Equation (1), then, has the form 

Xdx + X x 6x x + Ydy + Y x dy x = ... (3) 

Of the form F =. two equations here exist ; namely, 

(*x *)* +(j l yy *= : 

The equations DF= 0, accordingly, are 

xdx -f- y dy = 

(x t x) 6x. x (x x x)dx + O, y) 8y x 

O'i y) $y = to ) 

Here, two of the variations in (5) can be expressed 
in terms of the others and introduced in (3). Also for 



Fig. 232. 



(4) 



(*j 



FORMAL DEVELOPMENT. 471 

purposes of elimination Lagrange employed a per- Lagrange's 

indetermi- 

fectlv uniform and systematic procedure, which maynatecoeffi- 

cicnts. 

be pursued quite mechanically, without reflection. We 
shall use it here. It consists in multiplying each of the 
equations (5) by an indeterminate coefficient A, //, and 
adding each in this form to (3). So doing, we obtain 

\X+lxii{x x x)\6x +[*, +/*(*! *)]<**i ) _ 
[Y+Xy-(y t y)-]dy +[K,+ p{y x y)-\6y x f _ "" 

The coefficients of the four displacements may now 
be put directly = 0. For two displacements are ar- 
bitrary, and the two remaining coefficients may be 
made equal to zero by the appropriate choice of A and 
ft which is tantamount to an elimination of the two 
remaining displacements. 

We have, therefore, the four equations 

X -f- A x /< ( .v j x) 

X I + /<(*, .*) = 

We shall first assume that the coordinates are given, 
and seek the forces that maintain equilibrium. The 
values of A and // are each determined by equating to 
zero two coefficients. We get from the second and 
fourth equations, 

M = S and /1 = *-, 

* t * yx y 

whence 

y,~y,-y C) 

that is to say, the total component force impressed at 
N has the direction MN. From the first and third 
equations we get 



(6) 



472 THE SCIENCE OF MECHANICS, 

Their en,- \ = - X +^ X *-*\ X= - V +^'*- y \ 

ploymentin x V 

the deter- 

the n finai n f anc ^ ^ rom these by simple reduction 

equation. vr . -.r 

X+ X x 

y+y;-y (8) 

that is to say, the resultant of the forces applied at M 
and N acts in the direction OM. * 

The four force-components are accordingly subject 
to only two conditions, (7) and (8). The problem, con- 
sequently, is an indeterminate one ; as it must be from 
the nature of the case ; for equilibrium does not depend 
upon the absolute magnitudes of the forces, but upon 
their directions and relations. 

If we assume that the forces are given and seek the 
four coordinates, we treat equations (6) in exactly the 
same manner. Only, we can now make use, in addi- 
tion, of equations (4). Accordingly, we have, upon the 
elimination of A and /j, equations (7) and (8) and two 
equations (4). From these the following, which fully 
solve the problem, are readily deduced 
a(X+X x ) 



x = 



y = 



V{X.+ X t y + {Y+Y x y* 

VTx+ x x y + (.y+ y x y 



* The mechanical interpretation of the indeterminate coefficients ?., fi may 
be shown as follows. Equations (6) express the equilibrium of tv/o/ree points 
on which in addition to X, Y, X lt K t other forces act which answer to the re- 
maining expressions and just destroy X, Y,X lt Y t . The point N, for example, 
is in equilibrium if -Vj^ is destroyed by a force pi [x t x), undetermined as yet 
in magnitude, and Y t by a force [i-{y\ y). This supplementary force is due 
to the constraints. Its direction is determined ; though its magnitude is not. 
If we call the angle which it makes with the axis of abscissas a, we shall have 

that is to say, the force due to the connections acts in the direction of *. 



FORMAL DEVELOPMEXT. 473 

aCX-rXj) bX x 



Character 
of the pres- 



V(X -r x t y + ( r-f Y x y Vx\ + Y\ , P T 

lem 



V(X A\ + (K+ K t ) l/JT| + K 

Simple as this example is, it is yet sufficient to give 
us a distinct idea of the character and significance of 
Lagrange's method. The mechanism of this method is 
excogitated once for all, and in its application to par- 
ticular cases scarcely any additional thinking is re- 
quired. The simplicity of the example here selected 
being such that it can be solved by a mere glance at 
the figure, we have, in our study of the method, the 
advantage of a ready verification at every step. 

6. We will now illustrate the application of equa- General 
tion (2), which is Lagrange's form of statement of the solution 
D'Alembert's principle. There is no problem when icai prob- 
the masses move quite independently of one another. 
Each mass yields to the forces applied to it ; the va- 
riations 6x, Sy, S z . . . . are wholly arbitrary, and each 
coefficient may be singly put = 0. For the motion of 
n masses we thus obtain 3* simul- 
taneous differential equations. 

But if equations of condition 
(-F=0) obtain between the coordi- 
nates, these equations will lead to 
others (DF= 0) between the dis- oL 
placements or variations. With the 

Fig. 233. 

latter we proceed exactly as in the 
application of equation (1). Only it must be noted 
here that the equations J?= must eventually be em- 
ployed in their undifferentiated as well as in their dif- 
ferentiated form, as will best be seen from the follow- 
ing example. 




474 THE SCIENCE OF MECHANICS. 

a dynam- A heavy material point m, lying in a vertical plane 

ical exam- ,_,,, . , .... . .. , 

pie. X Y, is free to move on a straight line, y = ax, inclined 

at an angle to the horizon. (Fig. 233.) Here equa- 
tion (2) becomes 

and, since X=0, and Y= mg, also 

*+(*+;&)*- w 

The place of i^= is taken by 

y = ax (10) 

and for DF we have 

dy = ad x. 
Equation (9), accordingly, since 6y drops out and 
6x is arbitrary, passes into the form 
4*x ( d*y\ ft 

7/T + ^ + ^iJ fl = - 

By the differentiation of (10), or (F = 0), we have 

d 2 y d 2 x 

~di* = a 7/2' 
and, consequently, 

S+'-H-SH (ii > 

Then, by the integration of (n), we obtain 
a /2 



1 -I- 2 

and 

a 2 /2 

1 -\- a- ^ 
where b and c are constants of integration, determined 
by the initial position and velocity of m. This result 
can also be easily found by the direct method. 



FORMAL DEVELOPMEXT. 475 

Some care is necessary in the application of equa- a modifica- 

n n T^1 1 t ' n * l ^ S 

tion (i) if F = contains the time, lhe procedure in example, 
such cases may be illustrated by the following example. 
Imagine in the preceding case the straight line on 
which /// descends to move vertically upwards with the 
acceleration y. We start again from equation (9) 

F= is here replaced by 

y = ax + y^ (12) 

To form DF= 0, we vary (12) only with respect to x 
and v, for we are concerned here only with the possible 
displacement of the system in its position at any given 
instant, and not with the displacement that actually 
takes place in time. We put, therefore, as in the pre- 
vious case, 

6y = aSx, 
and obtain, as before, 

+(">- as) 

But to get an equation in .r alone, we have, since x 
and y are connected in (13) by the actual motion, to 
differentiate (12) with respect to / and employ the re- 
sulting equation 

d 2 y d 2 x 

Jt 2 = U It 2 + Y 

for substitution in (13). In this way the equation 
is obtained, which, integrated, gives 



476 



THE SCIENCE OF MECHANICS. 



a 



t 2 



y 



r 



1 -f- a 2 

r2 0r+r) 



(i + Y)t +i>t + c 



a- 



l + a'< 



t* 



-J- a b t -\- ac. 



If a weightless body m lie on the moving straight 
line, we obtain these equations 

I 2 



y 



1 -f- a 2 
y / 2 



Y -o+bt+c 



2 



-(- a < / -j- c, 



m> 



a 



1 + a 2 2 

results which are readily understood, when we re- 
flect that, on a straight line moving upwards with the 
acceleration y, m behaves as if it were affected with a 
downward acceleration y on the straight line at rest. 
Discussion 7- The procedure with equation (12) in the preced- 
itied exam- ing example may be rendered somewhat clearer by the 
following consideration. Equation (2), D'Alembert's 
principle, asserts, that all the work 
that can be done in the displacement 
of a system is done by the impressed 
forces and not by the connections. This 
is evident, since the rigidity of the con- 
nections allows no changes in the rela- 
tive positions which would be neces- 
sary for any alteration in the potentials of the elastic 
forces. But this ceases to be true when the connec- 
tions undergo changes in time. In this case, the changes 
of the connections perform work, and we can then ap- 
ply equation (2) to the displacements that actually take 
place only provided we add to the impressed forces the 
forces that produce the changes of the connections. 

A heavy mass m is free to move on a straight line 
parallel to OY (Fig. 234.) Let this line be subject to 



1 



X 



Fig. 234. 



FORMAL DEVELOPMEXT. 



477 



a forced acceleration in the direction of x, such that illustration 

of the mod- 

the equation F= becomes ifiedexam- 

^ pie. 

*~Y% ( 14 ) 

D'Alembert's principle again gives equation (9). 
But since from DF= it follows here that 6x = 0, 
this equation reduces itself to 

' g + l 2 Asj = o 

in which 6y is wholly arbitrary. Wherefore, 

,r-y 



(15) 



p- 4- --' = 



ind 



y 



<r/2 



-f at -\- b, 



to which must be supplied (14) or 

I 2 

It is patent that (15) does not' assign the total work 
of the displacement that actually takes place, but only 
that of some possible displacement on the straight line 
conceived, for the moment, as fixed. 

If we imagine the straight line massless, and cause 
it to travel parallel to itself in some guiding mechan- 
ism moved by a force my, equation (2) will be re- 
placed by 



a"* x 
my m )6x-\ 



*Tt^* y 



0. 



and since 6x, Sy are wholly arbitrary here, we obtain 
the two equations 

_d 2 x 
Y ~~dl* 



= 



478 THE SCIENCE OF MECHANICS. 

which give the same results as before. The apparently 
different mode of treatment of these cases is simply the 
result of a slight inconsistency, springing from the fact 
that all the forces involved are, for reasons facilitating 
calculation, not included in the consideration at the 
outset, but a portion is left to be dealt with subse- 
quently. 
Deduction 8. As the different mechanical principles only ex- 
cipie e oF"/i press different aspects of the same fact, any one of 
Lagrange's them is easily deducible from any other ; as we shall 
taTdynam- now illustrate by developing the principle of vis viva 
tion. equa from equation (2) of page 468. Equation (2) refers to 
instantaneously possible displacements, that is, to "vir- 
tual " displacements. But when the connections of a 
system are independent of the time, the motions that 
actually take place are "virtual" displacements. Conse- 
quently the principle may be applied to actual motions. 
For dx, Sy, dz, we may, accordingly, write dx, dy, 
dz, the displacements which take place in time, and 
put 

2 (Xdx + Ydy -f Zdz) = 

The expression to the right may,' by introducing for 
dx, (dx/dt) dl and so forth, and by denoting the velo- 
city by v, also be written 



fd 2 x dx , , d 2 y dy _, , d 2 z dz ,\ 



%d2m 



dX \ 2 JL. ( d A 2 '-L. [ d% 

Tt) ^\Tt) + \Tt, 



= Xd2mv 2 . 



FORMAL DEVELOPMENT. 479 

Also in the expression to the left, (dx/df) dt may be Force- 

r , ... function. 

written for ax. But this gives 

J2 (Xdx + Ydy + Zdz) = 2\m (p* - r g) f 

where v denotes the velocity at the beginning and v 
the velocity at the end of the motion. The integral to the 
left can always be found if we can reduce it to a single 
variable, that is to say, if we know the course of the 
motion in time or the paths which the movable points 
describe. If, however, X, Y, Z are the partial differ- 
ential coefficients of the same function Uoi coordinates, 
if, that is to say, 

Y _dU v _dU dU 

A -~dx'' ~dp Z ~7P 

as is always the case when only central forces are in- 
volved, this reduction is unnecessary. The entire ex- 
pression to the left is then a complete differential. And 
we have 

2((/-C/ ) = 2$ m (v*-vl), 

which is to say, the difference of the force-functions 
(or work) at the beginning and the end of the motion 
is equal to the difference of the vires viva at the be- 
ginning and the end of the motion. The vires viva are 
in such case also functions of the coordinates. 

In the case of a body movable in the plane of X 
and Ksuppose; for example, X= y, Y= jc; we 
then have 

J{-ydx - xdy) = -Jd{xy) = 

a V'o - v - v = \ m 3 r o> 

But if A' = a, Y= x, the integral to the left is 
f{a dx -\- x dy). This integral can be assigned the 
moment we know the path the body has traversed, that 



;8o 



THE SCIENCE OF MECHANICS. 



Essential 
character 
of analyt- 
ical me- 
chanics. 



is, if v is determined a function of x. If, for example, 
y =flx 2 , the integral would become 

2 s> (x x\ 3 

C{a + 2/*2) dx = a (x Q x) + 7 ^ J -. 

The difference of these two cases is, that in the first 
the work is simply a function of coordinates, that a 
force-function exists, that the element of the work is a 
complete differential, and the work consequently is de- 
termined by the initial and final values of the coordi- 
nates, while in the second case it is dependent on the 
entire path described. 

9. These simple examples, in themselves present- 
ing no difficulties, will doubtless suffice to illustrate the 
general nature of the operations of analytical mechan- 
ics. No fundamental light can be expected from this 
branch of mechanics. On the contrary, the discovery 
of matters of principle must be substantially completed 
before we can think of framing analytical mechanics ; 
the sole aim of which is a perfect" practical mastery of 
problems. Whosoever mistakes this situation, will 
never comprehend Lagrange's great performance, which 
here too is essentially of an economical character. Poin- 
sot did not altogether escape this error. 

It remains to be mentioned that as the result of the 
labors of Mobius, Hamilton, Grassmann, and others, a 
new transformation of mechanics is preparing. These 
inquirers have developed mathematical conceptions 
that conform more exactly and directly to our geomet- 
rical ideas than do the conceptions of common analyt- 
ical geometry ; and the advantages of analytical gene- 
rality and direct geometrical insight are thus united. 
But this transformation, of course, lies, as yet, beyond 
the limits of an historical exposition. (See p. 577.) 



FORMAL DEVELOPMEXT 4 tii 



THE ECONOMY OF SCIENCE. 

l. It is the object of scrence to replace, or save, ex- The basis 

, . , . .... of science. 

penences, by the reproduction and anticipation ot facts economy o 
in thought. Memory is handier than experience, and 
often answers the same purpose. This economical 
office of science, which fills its whole life, is apparent 
at first glance ; and with its full recognition all mys- 
ticism in science disappears. 

Science is communicated by instruction, in order 
that one man may profit by the experience of another 
and be spared the trouble of accumulating it for him- 
self ; and thus, to spare posterity, the experiences of 
whole generations are stored up in libraries. 

Language, the instrument of this communication, The eco- 
is itself an economical contrivance. Experiences are character 
analysed, or broken up, into simpler and more familiar gaage! 
experiences, and then symbolised at some sacrifice of 
precision. The symbols of speech are as yet restricted 
in their use within national boundaries, and doubtless 
will long remain so. But written language is gradually 
being metamorphosed into an ideal universal character. 
It is certainly no longer a mere transcript of speech. 
Numerals, algebraic signs, chemical symbols, musical 
notes, phonetic alphabets, may be regarded as parts 
already formed of this universal character of the fu- 
ture ; they are, to some extent, decidedly conceptual, 
and of almost general international use. The analysis 
of colors, physical and physiological, is already far 
enough advanced to render an international system of 
color-signs perfectly practical. In Chinese writing, 



482 THE SCIENCE OF MECHANICS. 

Possibility we have an actual example of a true ideographic Ian 

of a univer- o x- 

sal lan- guage, pronounced diversely in different provinces, yet 
everywhere carrying the same meaning. Were the 
system and its signs only of a simpler character, the 
use of Chinese writing might become universal. The 
dropping of unmeaning and needless accidents of gram- 
mar, as English mostly drops them, would be quite 
requisite to the adoption of such a system. But uni- 
versality would not be the sole merit of such a char- 
acter ; since to read it would be to understand it. Our 
children often read what they do not understand ; but 
that which a Chinaman cannot understand, he is pre- 
cluded from reading. 
Econom- 2. In the reproduction of facts in thought, we 

ler of au ac " never reproduce the facts in full, but only that side of 
Bematfom them which is important to us, moved to this directly 
^or l (. or indirectly by a practical interest. Our reproductions 
are invariably abstractions. Here again is an econom- 
ical tendency. 

Nature is composed of sensations as its elements. 
Primitive man, however, first picks out certain com- 
pounds of these elements those namely that are re- 
latively permanent and of greater importance to him. 
The first and oldest words are names of "things." 
Even here, there is an abstractive process, an abstrac- 
tion from the surroundings of the things, and from the 
continual small changes which these compound sensa- 
tions undergo, which being practically unimportant are 
not noticed. No inalterable thing exists. The thing 
is an abstraction, the name a symbol, for a compound 
of elements from whose changes we abstract. The 
reason we assign a single word to a whole compound is 
that we need to suggest all the constituent sensations 
at once. When, later, we come to remark the change- 



FORMAL DEVELOPMEXT 483 

ableness, we cannot at the same time hold fast to the 
idea of the thing's permanence, unless we have recourse 
to the conception of a thing-in-itself, or other such like 
absurdity. Sensations are not signs of things ; but, on 
the contrary, a thing is a thought-symbol for a com- 
pound sensation of relative fixedness. Properly speak- 
ing the world is not composed of "things "'as its ele- 
ments, but of colors, tones, pressures, spaces, times, 
in short what we ordinarily call individual sensations. 

The whole operation is a mere affair of economy. 
In the reproduction of facts, we begin with the more 
durable and familiar compounds, and supplement these 
later with the unusual by way of corrections. Thus, 
we speak of a perforated cylinder, of a cube with bev- 
eled edges, expressions involving contradictions, un- 
less we accept the view here taken. All judgments are 
such amplifications and corrections of ideas already 
admitted. 

3. In speaking of cause and effect we arbitrarily The ideas 
give relief to those elements to whose connection we effect, 
have to attend in the reproduction of a fact in the re- 
spect in which it is important to us. There is no cause 
nor effect in nature ; nature has but an individual exis- 
tence ; nature simply is. Recurrences of like cases in 
which A is always connected with B, that is, like results 
under iike circumstances, that is again, the essence of the 
connection of cause and effect, exist but in the abstrac- 
tion which we perform for the purpose of mentally re- 
producing the facts. Let a fact become familiar, and 
v/e no longer require this putting into relief of its con- 
necting marks, one attention is no longer attracted to 
the new and surprising, and we cease to speak of cause 
and effect. Heat is said to be the cause of the tension 
of steam ; but when the phenomenon becomes familiar 



484 THE SCIENCE OF MECHANICS. 

we think of the steam at once with the tension proper 
to its temperature. Acid is said to be the cause of the 
reddening of tincture of litmus ; but later we think of 
the reddening as a property of the acid. 
Hume, Hume first propounded the question, How can a 

Kant, and . r tr m i 

Schopen- thing A act on another thing B? Hume, in fact, re- 

hauer's ex- _ 

pianations jects causality and recognises only a wonted succes- 

of cause ... 

and eflect. sion in time. Kant correctly remarked that a necessary 
connection between A and B could not be disclosed by 
simple observation. He assumes an innate idea or 
category of the mind, a Verstandcsbegriff, under which 
the cases of experience are subsumed. Schopenhauer, 
who adopts substantially the same position, distin- 
guishes four forms of the "principle of sufficient rea- 
son" the logical, physical, and mathematical form, 
and the law of motivation. But these forms differ only 
as regards the matter to which they are applied, which 
may belong either to outward or inward experience. 
Cause and The natural and common-sense explanation is ap- 
economicai parently this. The ideas of cause and effect originally 
of thought, sprang from an endeavor to reproduce facts in thought. 
At first, the connection of A and B, of C and D, of E 
and E, and so forth, is regarded as familiar. But after 
a greater range of experience is acquired and a con- 
nection between M and N is observed, it often turns 
out that we recognise M as ?nade up of A, C, E, and N 
of B, D, E, the connection of which was before a fa- 
miliar fact and accordingly possesses with us a higher 
authority. This explains why a person of experience 
regards a new event with different eyes than the nov- 
ice. The new experience is illuminated by the mass 
of old experience. As a fact, then, there really does 
exist in the mind an "idea " under which fresh experi- 
ences are subsumed ; but that idea has itself been de- 






FORMAL DEVELOPMEXT. 485 

veloped from experience. The notion of the necessity 
of the causal connection is probably created by our 
voluntary movements in the world and by the changes 
which these indirectly produce, as Hume supposed but 
Schopenhauer contested. Much of the authority of 
the ideas of cause and effect is due to the fact that they 
are developed instinctively and involuntarily, and that 
we are distinctly sensible of having personally con- 
tributed nothing to their formation. We may, indeed, 
say, that our sense of causality is not acquired by the 
individual, but has been perfected in the develop- 
ment of the race. Cause and effect, therefore, are 
things of thought, having an economical office. It can- 
not be said why they arise. For it is precisely by the 
abstraction of uniformities that we know the question 
"why." (See Appendix, XXYI, p. 579.) 

4. In the details of science, its economical character Econom- 
ical fea- 
is still more apparent. The so-called descriptive sci- tures of 

, . n . . aI1 laws of 

ences must chiefly remain content with reconstructing nature, 
individual facts. Where it is possible, the common fea- 
tures of many facts are once for all placed in relief. But 
in sciences that are more highly developed, rules for the 
reconstruction of great numbers of facts may be embod- 
ied in a single expression. Thus, instead of noting indi- 
vidual cases of light-refraction, we can mentally recon- 
struct all present and future cases, if we know that the 
incident ray, the refracted ray, and the perpendicular 
lie in the same plane and that sin a sin /?= n. Here, 
instead of the numberless cases of refraction in different 
combinations of matter and under all different angles 
of incidence, we have simply to note the rule above 
stated and the values of //, which is much easier. The 
economical purpose is here unmistakable. In nature 
there is no law of refraction, only different cases of re- 



486 THE SCIENCE OF MECHANICS. 

fraction. The law of refraction is a concise compen- 
dious rule, devised by us for the mental reconstruction 
of a fact, and only for its reconstruction in part, that 
is, on its geometrical side. 
The econ- 5. The sciences most highly developed economically 

omy of the , ' , 

mathemat- are.those whose facts are reducible to a few numerable 

ical sci- _ _ , 

ences. elements of like nature. Such is the science of mechan- 
ics, in which we deal exclusively with spaces, times, 
and masses. The whole previously established econ- 
omy of mathematics stands these sciences in stead. 
Mathematics may be defined as the economy of count- 
ing. Numbers are arrangement-signs which, for the 
sake of perspicuity and economy, are themselves ar- 
ranged in a simple system. Numerical operations, it 
is found, are independent of the kind of objects operated 
on, and are consequently mastered once for all. When, 
for the first time, I have occasion to add five objects to 
seven others, I count the whole collection through, at 
once ; but when I afterwards discover that I can start 
counting from 5, I save myself part of the trouble ; 
and still later, remembering that 5 and 7 always count 
up to 12, I dispense with the numeration entirely. 
Arithmetic The object of all arithmetical operations is to save 
bra. a8e direct numeration, by utilising the results of our old 
operations of counting. Our endeavor is, having done 
a sum once, to preserve the answer for future use. The 
first four rules of arithmetic well illustrate this view. 
Such, too, is the purpose of algebra, which, substitut- 
ing relations for values, symbolises and definitively 
fixes all numerical operations that follow the same rule. 
For example, we learn from the equation 

x 2 y 2 

-<- = x y, 

x+y 



FORMAL DEVELOPMEXT. 487 

that the more complicated numerical operation at the 
left may always be replaced by the simpler one at the 
right, whatever numbers x and v stand for. We thus 
save ourselves the labor of performing in future cases 
the more complicated operation. Mathematics is the 
method of replacing in the most comprehensive and 
economical manner possible, nerv numerical operations 
by old ones done already with known results. It may 
happen in this procedure that the results of operations 
are employed which were originally performed centu- 
ries ago. 

Often operations involving intense mental effort The theory 

. . .of deter- 

may be replaced by the action of semi-mechanical miiwnw . 
routine, with great saving of time and avoidance of 
fatigue. For example, the theory of determinants 
owes its origin to the remark, that it is not necessary 
to solve each time anew equations of the form 

a, x+ 6 t v -J- c x = 

a x-f o 2 y + <r 2 = 0, 
from which result 



1 
1 


-a x b x 


P 

N 


1 
*1 


c, </._, c __ 

h a 2 b l 


Q 



7 = 

but that the solution may be effected by means of the 
coefficients, by writing down the coefficients according 
to a prescribed scheme and operating with them me- 
chanically. Thus, 



a t l>, 

a., Ik 



a i b 2 a 2 b \ = N 



and similarly 



f 1 =P, and * C ' = Q. 

l>* a n < ^ 



488 THE SCIENCE OF MECHANICS. 

Calculating Even a total disburdening of the mind can be ef- 

niachines. , _ _ 

fected in mathematical operations. This happens where 
operations of counting hitherto performed are symbol- 
ised by mechanical operations with signs, and our brain 
energy, instead of being wasted on the repetition of 
old operations, is spared for more important tasks. 
The merchant pursues a like economy, when, instead 
of directly handling his bales of goods, he operates 
with bills of lading or assignments of them. The 
drudgery of computation may even be relegated to a 
machine. Several different types of calculating ma- 
chines are actually in practical use. The earliest of 
these (of any complexity) was the difference-engine of 
Babbage, who was familiar with the ideas here pre- 
sented, 
other ab- A numerical result is not always reached by the 

methods of actual solution of the problem ; it may also be reached 

attaining . , . , T . . . 

results. indirectly. It is easy to ascertain, for example, that a 
curve whose quadrature for the abscissa x has the value 
x'", gives an increment mx'"~ I dx of the quadrature for 
the increment dx of the abscissa. But we then also know 
that Cmx m ~ l dx = x m ; that is, we recognise the quan- 
tity x" 1 from the increment mx m ~ x dx as unmistakably 
as we recognise a fruit by its rind. Results of this 
kind, accidentally found by simple inversion, or by 
processes more or less analogous, are very extensively 
employed in mathematics. 

That scientific work should be more useful the more 
it has been used, while mechanical work is expended in 
use, may seem strange to us. When a person who 
daily takes the same walk accidentally finds a shorter 
cut, and thereafter, remembering that it is shorter, al- 
ways goes that way, he undoubtedly saves himself the 
difference of the work. But memory is really not work. 



FORMAL DEVELOPMEXT. 489 

It only places at our disposal energy within our present 
or future possession, which the circumstance of igno- 
rance prevented us from availing ourselves of. This 
is precisely the case with the application of scientific 
ideas. 

The mathematician who pursues his studies with- Necessity 

of clear 

out clear views of this matter, must often have the views on 

this sub- 
uncomfortable feeling that his paper and pencil sur- ject. 

pass him in intelligence. Mathematics, thus pursued 
as an object of instruction, is scarcely of more educa- 
tional value than busying oneself with the Cabala. On 
the contrary, it induces a tendency toward mystery, 
which is pretty sure to bear its fruits. 

6. The science of physics also furnishes examples Example* 
of this economy of thought, altogether similar to those omy of 
we have just examined. A brief reference here will suf- physics, 
fice. The moment of inertia saves us the separate con- 
sideration of the individual particles of masses. By 
the force-function we dispense with the separate in- 
vestigation of individual force-components. The sim- 
plicity of reasonings involving force-functions springs 
from the fact that a great amount of mental work had 
to be performed before the discovery of the properties 
of the force-functions was possible. Gauss's dioptrics 
dispenses us from the separate consideration of the 
single refracting surfaces of a dioptrical system and 
substitutes for it the principal and nodal points. But 
a careful consideration of the single surfaces had to 
precede the discovery of the principal and nodal points. 
Gauss's dioptrics simply saves us the necessity of often 
repeating this consideration. 

We must admit, therefore, that there is no result of 
science which in point of principle could not have been 
arrived at wholly without methods. But, as a matter 



49Q THE SCIENCE OF MECHANICS. 

Science a of fact, within the short span of a human life and with 

minimal . 

problem, man's limited powers of memory, any stock of knowl- 
edge worthy of the name is unattainable except by the 
greatest mental economy. Science itself, therefore, 
may be regarded as a minimal problem, consisting of 
the completest possible presentment of facts with the 
least possible expenditure of thought. 

7. The function of science, as we take it, is to re- 
place experience. Thus, on the one hand, science 
must remain in the province of experience, but, on the 
other, must hasten beyond it, constantly expecting con- 
firmation, constantly expecting the reverse. Where 
neither confirmation nor refutation is possible, science 
is not concerned. Science acts and only acts in the 
domain of uncompleted experience. Exemplars of such 
branches of science are the theories of elasticity and 
of the conduction of heat, both of which ascribe to the 
smallest particles of matter only such properties as ob- 
servation supplies in the study of the larger portions. 
The comparison of theory and experience may be far- 
ther and farther extended, as our means of observation 
increase in refinement. 

The princi- Experience alone, without the ideas that are asso- 

ple of con- ..... 1 i r 

tinuity, the ciated with it, would forever remain strange to us. 

entific Those ideas that hold good throughout the widest do- 
mains of research and that supplement the greatest 
amount of experience, are the most scientific. The prin- 
ciple of continuity, the use of which everywhere per- 
vades modern inquiry, simply prescribes a mode of 
conception which conduces in the highest degree to the 
economy of thought. 

8. If a long elastic rod be fastened in a vise, the 
rod may be made to execute slow vibrations. These 
are directly observable, can be seen, touched, and 



method. 



FORMAL DEVELOPMEXT. 491 

graphically recorded. If the rod be shortened, the Example 3- 

... ... . . ... , .1- lustrative 

vibrations will increase in rapidity and cannot be di- of the 
rectly seen ; the rod will present to the sight a blurred science, 
image. This is a new phenomenon. But the sensa- 
tion of touch is still like that of the previous case ; we 
can still make the rod record its movements ; and if 
we mentally retain the conception of vibrations, we can 
still anticipate the results of experiments. On further 
shortening the rod the sensation of touch is altered ; 
the rod begins to sound ; again a new phenomenon is 
presented. But the phenomena do not all change at 
once; only this or that phenomenon changes; conse- 
quently the accompanying notion of vibration, which 
is not confined to any single one, is still serviceable, 
still economical. Even when the sound has reached 
so high a pitch and the vibrations have become so 
small that the previous means of observation are not 
of avail, we still advantageously imagine the sounding 
rod to perform vibrations, and can predict the vibra- 
tions of the dark lines in the spectrum of the polarised 
light of a rod of glass. If on the rod being further 
shortened all the phenomena suddenly passed into new 
phenomena, the conception of vibration would no 
longer be serviceable because it would no longer afford 
us a means of supplementing the new experiences by 
the previous ones. 

When we mentally add to those actions of a human 
being which we can perceive, sensations and ideas like 
our own which we cannot perceive, the object of the 
idea we so form is economical. The idea makes ex- 
perience intelligible to us ; it supplements and sup- 
plants experience. This idea is not regarded as a great 
scientific discovery, only because its formation is so 
natural that every child conceives it. Now, this is 



492 THE SCIENCE OF MECHANICS 

exactly what we do when we imagine a moving body 
which has just disappeared behind a pillar, or a comet 
at the moment invisible, as continuing its motion and 
retaining its previously observed properties. We do 
this that we may not be surprised by its reappearance. 
We fill out the gaps in experience by the ideas that 
experience suggests. 
aii scien- 9- Yet not all the prevalent scientific theories origi- 

ories not nated so naturally and artlessly. Thus, chemical, elec- 
the princi- trical, and optical phenomena are explained by atoms. 
finuity? on But the mental artifice atom was not formed by the 
principle of continuity ; on the contrary, it is a pro- 
duct especially devised for the purpose in view. Atoms 
cannot be perceived by the senses ; like all substances, 
they are things of thought. Furthermore, the atoms 
are invested with properties that absolutely contradict 
the attributes hitherto observed in bodies. However 
well fitted atomic theories may be to reproduce certain 
groups of facts, the physical inquirer who has laid to 
heart Newton's rules will only admit those theories as 
provisional helps, and will strive to attain, in some more 
natural way, a satisfactory substitute. 
Atoms and The atomic theory plays a part in physics similar 

other men- . - . ... ... 

tai artifices, to that of certain auxiliary concep ts in mathematics; 
it is a mathematical model for facilitating the mental 
reproduction of facts. Although we represent vibra- 
tions by the harmonic formula, the phenomena of cool- 
ing by exponentials, falls by squares of times, etc., no 
one will fancy that vibrations in themselves have any- 
thing to do with the circular functions, or the motion 
of falling bodies with squares. It has simply been ob- 
served that the relations between the quantities inves- 
tigated were similar to certain relations obtaining be- 
tween familiar mathematical functions, and these more 



FORMAL DEVELOPMEXT. 493 

familiar ideas are employed as an easy means of sup- 
plementing experience. Natural phenomena whose re- 
lations are not similar to those of functions with which 
we are familiar, are at present very difficult to recon- 
struct. But the progress of mathematics may facilitate 
the matter. 

As mathematical helps of this kind, spaces of more Muiti- 

dimen- 

than three dimensions may be used, as I have else- sioned 

spaces 

where shown. But it is not necessary to regard these, 
on this account, as anything more than mental arti- 
fices.* 

As the outcome of the labors of Lobatchevski, Bolyai, Gauss, and Rie- 
mann, the view has gradually obtained currency in the mathematical world, 
that that which we call space is a particular, actual case of a more general, 
conceivable case of multiple quantitative manifoldness. The space of sight 
and touch is a threefold manifoldness ; it possesses three dimensions ; and 
every point in it can be defined by three distinct and independent data. But 
it is possible to conceive of a quadruple or even multiple space-like manifold- 
ness. And the character of the manifoldness may also be differently conceived 
from the manifoldness of actual space. We regard this discovery, which is 
chiefly due to the labors of Riemann, as a very important one. The properties 
of actual space are here directly exhibited as objects of experience, and the 
pseudo-theories of geometry that seek to excogitate these properties by meta- 
physical arguments are overthrown. 

A thinking being is supposed to live in the surface of a sphere, with no 
other kind of space to institute comparisons with. His space will appear to 
him similarly constituted throughout. He might regard jt as infinite, and 
could only be convinced of the contrary by experience. Starting from any two 
points of a great circle of the sphere and proceeding at right angles thereto on 
other great circles, he could hardly expect that the circles last mentioned 
would intersect. So, also, with respect to the space in which we live, only ex- 
perience can decide whether it is finite, whether parallel lines intersect in it, 
or the like. The significance of this elucidation can scarcely be overrated. 
An enlightenment similar to that which Riemann inaugurated in science was 
produced in the mind of humanity at large, as regards the surface of the earth, 
by the discoveries of the first circumnavigators. 

The theoretical investigation of the mathematical possibilities above re- 
ferred to, has, primarily, nothing to do with the question whether things really 
exist which correspond to these possibilities; and we must not hold mathe- 
maticians responsible for the popular absurdities which their investigations 
have given rise to. The space of sight and touch is Mr<r-dimensionaI ; that, 
no one ever yet doubted. If, now, it should be found that bodies vanish from 
this space, or new bodies get into it, the question might scientifically be dis- 
cussed whether it would facilitate and promote our insight into things to con- 
ceive experiential space as part of a four-dimensional or multi-dimensional 



494 THE SCIENCE OF MECHANICS. 

Hypotheses This is the case, too, with all hypothesis formed 

and facts . 

for tne explanation of new phenomena. Our concep- 
tions of electricity fit in at once with the electrical phe- 
nomena, and take almost spontaneously the familiar 
course, the moment we note that things take place as 
if attracting and repelling fluids moved on the surface 
of the conductors. But these mental expedients' have 
nothing whatever to do with the phenomenon itself. 
(See Appendix, XXVII, p. 579.) 

space. Yet in such a case, this fourth dimension would, none the less, remain 
a pure thing of thought a mental fiction. 

But this is not the way matters stand. The phenomena mentioned were 
not forthcoming until after the new views were published, and were then ex- 
hibited in the presence of certain persons at spiritualistic seances. The fourth 
dimension was a very opportune discover'- for the spiritualists and for theo- 
logians who were in a quandary about the location of hell. The use the spiri- 
tualist makes of the fourth dimension is this. It is possible to move out of a 
finite straight line, without passing the extremities, through the second dimen- 
sion ; out of a finite closed surface through the third ; and, analogously, out 
of a finite closed space, without passing through the enclosing boundaries, 
through the fourth dimension. Even the tricks that prestidigitateurs, in the 
old days, harmlessly executed in three dimensions, are now invested with a 
new halo by the fourth. But the tricks of the spiritualists, the tying or untying 
of knots in endless strings, the removing of bodies from closed spaces, are all 
performed in cases where there is absolutely nothing at stake. All is purpose- 
less jugglery. We have not yet found an accoucheur who has accomplished 
parturition through the fourth dimension. If we should, the question would 
at once become a serious one. Professor Simony's beautiful tricks in rope- 
tying, which, as the performance of a prestidigitateur, are very admirable, 
speak against, not for, the spiritualists. 

Everyone is free to set up an opinion and to adduce proofs in support of 
it. Whether, though, a scientist shall find it worth his while to enter into 
serious investigations of opinions so advanced, is a question which his reason 
and instinct alone can decide. If these things, in the end, should turn out to 
be true, I shall not be ashamed of being the last to believe them. What I have 
seen of them was not calculated to make me less sceptical. 

I myself regarded multi-dimensioned spare as a mathematico-physical 
help even prior to the appearance of Niemann's memoir. But I trust that 
no one will employ what I have thought, said, and written on this subject as a 
basis for the fabrication of ghost stories. (Compare Mach, Die Ceschi ohte und 
die IVurzel des Satzes von der ErhaLtun* der Arbeit.) 

AM mU y*iy - s^f . |>.49 ^ 



CHAPTER V. 

THE RELATIONS OF MECHANICS TO OTHER DE- 
PARTMENTS OF KNOWLEDGE, 
i. 
THE RELATIONS OF MECHANICS TO PHYSICS. 

1. Purely mechanical phenomena do not exist. The T , he events 

J r of nature 

woduction of mutual accelerations in masses is, to all d p n . ot ?*- 

clusively 

ippearances, a purely dynamical phenomenon. But belon R. to 

nth these d)'namical results are always associated ence - 

lermal, magnetic, electrical, and chemical phenom- 

la, and the former are always modified in proportion 

is the latter are asserted. On the other hand, thermal, 

lagnetic, electrical, and chemical conditions also can 

jroduce motions. Purely mechanical phenomena, ac- 

:ordingly, are abstractions, made, either intentionally 

)r from necessity, for facilitating our comprehension of 

lings. The same thing is true of the other classes of 

)hysical phenomena. Every event belongs, in a strict 

mse, to all the departments of physics, the latter be- 

lg separated only by an artificial classification, which 

partly conventional, partly physiological, and partly 

listcrical. 

2. The view that makes mechanics the basis of the 
remaining branches of physics, and explains all physical 
)henomena by mechanical ideas, is in our judgment a 
>rejudice. Knowledge which is historically first, is 
lot necessarily the foundation of all that is subsequently 



496 THE SCIENCE OF MECHANICS. 

Tin me- gained. As more and more facts are discovered and 

chanical 

aspects of classified, entirely new ideas of general scope can be 

nature not . J . 

necessarily formed. We have no means of knowing, as vet, which 

its funda- " < . ' i ' 

mental of the physical phenomena go deepest, whether the 
mechanical phenomena are perhaps not the most super- 
ficial of all, or whether all do not go equally deep. Even 
in mechanics we no longer regard the oldest law, the 
law of the lever, as the foundation of all the ether 
principles. 

Artificiality The mechanical theory of nature, is, undoubtedly, 

of the me- . ... ... 

chanical in an historical view, both intelligible and pardonable ; 

conception . . 

of the and it may also, for a time, have been of much value. 

world. J ' '. 

But, upon the whole, it is an artificial conception. 
Faithful adherence to the method that led the greatest 
investigators of nature, Galileo, Newton, Sadi Carnot, 
Faraday, and J. R. Mayer, to their great results, re- 
stricts physics to the expression of actual facts, and 
forbids the construction of hypotheses behind the facts, 
where nothing tangible and verifiable is found. If this 
is done, only the simple connection of the motions of 
masses, of changes of temperature, of changes in the 
values of the potential function, of chemical changes, 
and so forth is to be ascertained, and nothing is to be 
imagined along with these elements except the physical 
attributes or characteristics directly or indirectly given 
by observation. 

This idea was elsewhere * developed by the author 
with respect to the phenomena of heat, and indicated, 
in the same place, with respect to electricity. All hy- 
potheses of fluids or media are eliminated from the 
theory of electricity as entirely superfluous, when we 
reflect that electrical conditions are all given by the 

* Mach, Die Geschichte unci die Wurzel des Satzes von der Erhaltung der 

Arbeit. 



See S4J- Y A? 



ITS RELATIOXS TO OTHER SCIEXCES. 497 

values of the potential function V and the dielectric Science 

should be 

constants. If we assume the differences of the values based on 

facts, not 

: j be measured (on the electrometer) by the forces, on hypoth- 
and regard Fand not the quantity of electricity Q as 
the primary notion, or measurable physical attribute, 
we shall have, for any simple insulator, for our quan- 
tity of electricity 

re x, v, z denote the coordinates and dv the ele- 
ment of volume,) and for our potential* 

8rr J V **' <b' <^ 3 J 
Here Q and W appear as derived notions, in which no 
conception of fluid or medium is contained. If we 
work over in a similar manner the entire domain of 
physics, we shall restrict ourselves wholly to the quan- 
titative conceptual expression of actual facts. All su- 
perfluous and futile notions are eliminated, and the 
imaginary problems to which they have given rise fore- 
ed. (See Appendix XXVIII, p. 583.) 
The removal of notions whose foundations are his- 
torical, conventional, or accidental, can best be fur- 
thered by a comparison of the conceptions obtaining 
in the different departments, and by finding for the 
conceptions of every department the corresponding 
conceptions of others. We discover, thus, that tem- 
peratures and potential functions correspond to the 
velocities of mass-motions. A single velocity- value, a 
single temperature-value, or a single value of potential 
function, never changes alone. But whilst in the case 
of velocities and potential functions, so far as we yet 

* Using the terminology of Clausius. 



498 



THE SCIENCE OF MECHANICS. 



Desirabii- know, only differences come into consideration, the 

ityofa . .1 , . 

compara- significance of temperature is not only contained in its 

tive phys- 

ics. difference with respect to other temperatures. Thermal 

capacities correspond to masses, the potential of an 
electric charge to quantity of heat, quantity of elec- 
tricity to entropy, and so on. The pursuit of such re- 
semblances and differences lays the foundation of a 
comparative physics, which shall ultimately render pos- 
sible the concise expression of extensive groups of facts, 
without arbitrary additions. We shall then possess a 
homogeneous physics, unmingled with artificial atomic 
theories. 

It will also be perceived, that a real economy of 
scientific thought cannot be attained by mechanical 
hypotheses. Even if an hypothesis were fully com- 
petent to reproduce a given department of natural phe- 
nomena, say, the phenomena of heat, we should, by 
accepting it, only substitute for the actual relations be- 
tween the mechanical and thermal processes, the hy- 
pothesis. The real fundamental facts are replaced by 
an equally large number of hypotheses, which is cer- 
tainly no gain. Once an hypothesis has facilitated, 
as best it can, our view of new facts, by the substitu- 
tion of more familiar ideas, its powers are exhausted. 
We err when we expect more enlightenment from an 
hypothesis than from the facts themselves. 

3. The development of the mechanical view was 
favored by many circumstances. In the first place, a 
connection of all natural events with mechanical pro- 

mechanica! cesses is unmistakable, and it is natural, therefore, that 
we should be led to explain less known phenomena by 
better known mechanical events. Then again, it was 
first in the department of mechanics that laws of gen- 
eral and extensive scope were discovered. A law of 



Circum- 
stances 
which fa- 
vored the 
develop 



ITS RELATIONS TO OTHER SCIENCES. 499 

this kind is the principle of vis viva '2(U' X Cf ) = 
2%m (rf v\), which states that the increase of the 
vis viva of a system in its passage from one position to 
another is equal to the increment of the force-function, 
or work, which is expressed as a function of the final 
and initial positions. If we fix our attention on the 
work a system can perform and call it with Helmholtz 
the Spannkrafl, S,* then the work actually performed, 
U, will appear as a diminution of the Spannkraft, K, 
initially present; accordingly, S=K U, and the 
principle of vis viva takes the form 

2S -\- ^ 2 m v- = const, 
that is to say, every diminution of the Spannkraft, isTheCon- 

J J 1 j servationof 

compensated for by an increase of the vis viva. In this Energy, 
form the principle is also called the law of the Conser- 
vation of Energy, in that the sum of the Spannkrafi (the 
potential energy) and the vis viva (the kinetic energy) 
remains constant in the system. But since, in nature, 
it is possible that not only vis viva should appear as the 
consequence of work performed, but also quantities of 
heat, or the potential of an electric charge, and so forth, 
scientists saw in this law the expression of a mechanical 
action as the basis of all natural actions. However, 
nothing is contained in the expression but the fact of 
an invariable quantitative connection between mechani- 
cal and other kinds of phenomena. 

4. It would be a mistake to suppose that a wide 
and extensive view of things was first introduced into 
physical science by mechanics. On the contrary, this 

* Helmholtz used this term in 1847; but it is not found in his subsequent 
papers; and in 1882 [Wissenschaftliche Abhandlungcn, II, 965) he expressly 
discards it in favor of the English " potential energy." He even (p. 968) pre- 
fers Clausius's word Ergal to Spannkraft, which is quite out of agreement 
witli modern terminology. Trans. 



500 THE SCIENCE OE MECHANICS. 

Compre- insight was possessed at all times by the foremost 

hensive- ... . 

nessof inquirers and even entered into the construction of 

view the ... 

condition, mechanics itself, and was, accordingly, not first created 

not the re- .. 

suit, of me- by the latter. Galileo and Huvgens constantly alter- 

chanics. . 

nated the consideration of particular details with the 
consideration of universal aspects, and reached their 
results only by a persistent effort after a simple and 
consistent view. The fact that the velocities of indi- 
vidual bodies and systems are dependent on the spaces 
descended through, was perceived by Galileo and 
Huygens only by a very detailed investigation of the 
motion of descent in particular cases, combined with 
the consideration of the circumstance that bodies gen- 
erally, of their own accord, only sink. Huygens 
especially speaks, on the occasion of this inquiry, of 
the impossibility of a mechanical perpetual motion ; 
he possessed, therefore, the modern point of view. He 
felt the incompatibility of the idea of a perpetual motion 
with the notions of the natural mechanical processes 
with which he was familiar. 
ExempHfi- Take the fictions of Stevinus say, that of the end- 

tiiis in ste- less chain on the prism. Here, too, a deep, broad 
searches? insight is displayed. We have here a mind, disciplined 
by a multitude of experiences, brought to bear on an 
individual case. The moving endless chain is to Ste- 
vinus a motion of descent that is not a descent, a mo- 
tion without a purpose, an intentional act that does 
not answer to the intention, an endeavor for a change 
which does not produce the change. If motion, gener- 
ally, is the result of descent, then in the particular case 
descent is the result of motion. It is a sense of the 
mutual interdependence of v and h in the equation 
v == \ / 2gh that is here displayed, though of course in 
not so definite a form. A contradiction exists in this 



ITS RELATIOXS TO OTHER SCIEXCES. 501 

fiction for Stevinus's exquisite investigative sense that 
would escape less profound thinkers. 

This same breadth of view, which alternates the a!so. in the 

1 - t j- 1 j 1 researches 

individual with the universal, is also displaved, only in of Camot 

... and J. R. 

this instance not restricted to mechanics, in the per- Mayer, 
formances of Sadi Carnot. When Carnot finds that 
the quantity of heat Q which, for a given amount of 
work Z, has flowed from a higher temperature /to a 
lower temperature t ', can only depend on the tempera- 
tures and not on the material constitution of the bodies, 
he reasons in exact conformity with the method of 
Galileo. Similarly does J. R. Mayer proceed in the 
enunciation of the principle of the equivalence of heat 
and work. In this achievement the mechanical view 
was quite remote from Mayer's mind ; nor had he need 
of it. They who require the crutch of the mechanical 
philosophy to understand the doctrine of the equiva- 
lence of heat and work, have only half comprehended 
the progress which it signalises. Yet, high as we may 
place Mayer's original achievement, it is not on that 
account necessary to depreciate the merits of the pro- 
fessional physicists Joule, Helmholtz, Clausius, and 
Thomson, who have done very much, perhaps all, to- 
wards the detailed establishment and perfection of the 
new view. The assumption of a plagiarism of Mayer's 
ideas is in our opinion gratuitous. They who advance 
it, are under the obligation to prove it. The repeated 
appearance of the same idea is not new in history. We 
shall not take up here the discussion of purely personal 
questions, which thirty years from now will no longer 
interest students. But it is unfair, from a pretense of 
justice, to insult men, who if they had accomplished 
but a third of their actual services, would have lived 
highly honored and unmolested lives. (See p. 584.) 



5 o2 THE SCIENCE OF MECHANICS. 

The inter- 5. We shall now attempt to show that the broad 

enceofthe view expressed in the principle of the conservation 

facts of na- . . 

ture. of energy, is not peculiar to mechanics, but is a condi- 

tion of logical and sound scientific thought generally. 
The business of physical science is the reconstruction 
of facts in thought, or the abstract quantitative expres- 
sion of facts. The rules which we form for these recon- 
structions are the laws of nature. In the conviction that 
such rules are possible lies the law of causality. The 
law of causality simply asserts that the phenomena of 
nature are dependent on one another. The special em- 
phasis put on space and time in the expression of the 
law of causality is unnecessary, since the relations of 
space and time themselves implicitly express that phe- 
nomena are dependent on one another. 

The laws of nature are equations between the meas- 
urable elements afiyd . . . . a? of phenomena. As na- 
ture is variable, the number of these equations is al- 
ways less than the number of the elements. 

If we know all the values of a fiyd . . ., by which, 
for example, the values of X fxv . . . are given, we may 
call the group afiyd. . . the cause and the group 
X/uv . . . the effect. In this sense we may say that the 
effect is uniquely determined by the cause. The prin- 
ciple of sufficient reason, in the form, for instance, in 
which Archimedes employed it in the development of 
the laws of the lever, consequently asserts nothing 
more than that the effect cannot by any given set of 
circumstances be at once determined and undetermined. 

If two circumstances a and X are connected, then, 
supposing all others are constant, a change of X will 
be accompanied by a change of a, and as a general 
rule a change of or by a change of A. The constant 
observance of this mutual interdependence is met with 



ITS KELATIOXS TO OTHER SCIENCES. 



53 




FiR. 235- 



in Stevinus, Galileo, Huygens, and other great inquir- Sense of 

. . . tn ' s inier- 

ers. The idea is also at the basis of the discovery of depend- 

ence at the 

counter- phenomena. Thus, a change in the volume of basis of ail 

... great dis- 

a gas due to a change of temperature is supplemented coveries. 
by the counter-phenomenon of a change of tempera- 
ture on an alteration of volume ; Seebeck's phenome- 
non by Peltier's effect, and so forth. 
Care must, of course, be exercised, in 
such inversions, respecting the form 
of the dependence. Figure 235 will 
render clear how a perceptible altera- 
tion of a may always be produced by 
an alteration of A, but a change of A 
not necessarily by a change of a. The relations be- 
tween electromagnetic and induction phenomena, dis- 
covered by Faraday, are a good instance of this truth. 

If a set of circumstances aftyS . . ., bv which a Various , 

' ' forms of ex- 

Second set Xuv . . . is determined, be made to pass pression ot 

/ * this truth. 

from its initial values to the terminal values a' fi'y 
6'. . ., then X^iv . . . also will pass into A' /*' v' . . . 
If the first set be brought back to its initial state, also 
the second set will be brought back to its initial state. 
This is the meaning of the "equivalence of cause and 
effect," which Mayer again and again emphasizes. 

If the first group suffer only periodical changes, the 
second group also can suffer only periodical changes, 
not continuous permanent ones. The fertile methods 
of thought of Galileo, Huygens, S. Carnot, Mayer, 
and their peers, are all reducible to the simple but sig- 
nificant perception, that purely periodical alterations of 
one set of circumstances can only constitute the source of 
similarly periodical alterations of a second set of circum- 
stances, not of continuous and permanent alterations. Such 
laxims, as "the effect is equivalent to the cause," 



5o 4 THE SCIENCE OF MECHANICS. 

"work cannot be created out of nothing," "a per- 
petual motion is impossible," are particular, less defi- 
nite, and less evident forms of this perception, which 
in itself is not especially concerned with mechanics, but 
is a constituent of scientific thought generally. With 
the perception of this truth, any metaphysical mystic- 
ism that may still adhere to the principle of the con- 
servation of energy* is dissipated. (See p. 585.) 
Purpose of All ideas of conservation, like the notion of sub- 

the ideas of 

conserva- stance, have a solid foundation in the economy of 

tion. ' 

thought. A mere unrelated change, without fixed point 
of support, or reference, is not comprehensible, not 
mentally reconstructible. We always inquire, accord- 
ingly, what idea can be retained amid all variations as 
permanent, what law prevails, what equation remains 
fulfilled, what quantitative values remain constant ? 
When we say the refractive index remains constant in 
all cases of refraction, ^remains = 9-810;// in all cases 
of the motion of heavy bodies, the energy remains con- 
stant in every isolated system, all our assertions have 
one and the same economical function, namely that of 
facilitating our mental reconstruction of facts. 

II. 
THE RELATIONS OF MECHANICS TO PHYSIOLOGY. 

Conditions I' All science has its origin in the needs of life. 

devetop? 6 However minutely it may be subdivided by particular 

science. vocations or by the restricted tempers and capacities of 

those who foster it, each branch can attain its full and 

best development only by a living connection with the 

whole. Through such a union alone can it approach 

* When we reflect that the principles of science are all abstractions that 
presuppose repetitions of similar cases, the absurd applications of the law of 
the conservation of forces to the universe as a whole fall to the ground. 



ITS RELATIOXS TO OTHER SCIEXCES. 505 

its true maturity, and be insured against lop-sided and 
monstrous growths. 

The division of labor, the restriction of individual Confusion 

.... , . . . of the 

inquirers to limited provinces, the investigation of means and 
those provinces as a life-work, are the fundamental science, 
conditions of a fruitful development of science. Only 
by such specialisation and restriction of work can the 
economical instruments of thought requisite for the 
mastery of a special field be perfected. But just here r 

lies a danger the danger of our overestimating the in- 
struments, with which we are so constantly employed, 
or even of regarding them as the objective point of 
science. 

2. Now, such a state of affairs has, in our opinion, physics 
actually been produced by the disproportionate formal madlfine 
development of physics. The majority of natural in physiology, 
quirers ascribe to the intellectual implements of physics, 

to the concepts mass, force, atom, and so forth, whose 
sole office is to revive economical!}' arranged expe- 
riences, a reality beyond and independent of thought. 
Not only so, but it has even been held that these forces 
and masses are the real objects of inquiry, and, if once 
they were fully explored, all the rest would follow from 
the equilibrium and motion of these masses. A person 
who knew the world only through the theatre, if brought 
behind the scenes and permitted to view the mechan- 
ism of the stage's action, might possibly believe that 
the real world also was in need of a machine-room, and 
that if this were once thoroughly explored, we should 
know all. Similarly, we, too, should beware lest the 
intellectual machinery, employed in the representation 
of the world on the stage of thought, be regarded as the 
basis of the real world. 

3. A philosophy is involved in any correct view of 



5 o6 THE SCIENCE OF MECHANICS. 

The at- the relations of special knowledge to the great body of 

tempt to ex- 

plain feel- knowledge at large, a philosophy that must be de- 

mgs by 

motions, manded of every special investigator. The lack of it 
is asserted in the formulation of imaginary problems, 
in the very enunciation of which, whether regarded as 
soluble or insoluble, flagrant absurdity is involved. 
Such an overestimation of physics, in contrast to physi- 
ology, such a mistaken conception of the true relations 
of the two sciences, is displayed in the inquiry whether 
it is possible to explain feelings by the motions of 
atoms? 

Explication Let us seek the conditions that could have impelled 

of this . . r 

anomaly, the mind to formulate so curious a question. We find 
in the first place that greater confidence is placed in our 
experiences concerning relations of time and space ; 
that we attribute to them a more objective, a more real 
character than to our experiences of colors, sounds, 
temperatures, and so forth. Yet, if we investigate the 
matter accurately, we must surely admit that our sen- 
sations of time and space are just as much sensations 
as are our sensations of colors, sounds, and odors, only 
that in our knowledge of the former we are surer and 
clearer than in that of the latter. Space and time are 
well-ordered systems of sets of sensations. The quan- 
tities stated in mechanical equations are simply ordinal 
symbols, representing those members of these sets 
that are to be mentally isolated and emphasised. The 
equations express the form of interdependence of these 
ordinal symbols. 

A body is a relatively constant sum of touch and 
sight sensations associated with the same space and 
time sensations. Mechanical principles, like that, for 
instance, of the mutually induced accelerations of two 
masses, give, either directly or indirectly, only some 



ITS RELATIONS TO OTHER SCIENCES . 507 

combination of touch, sight, light, and time sensations. 
They possess intelligible meaning only by virtue of 
the sensations they involve, the contents of which may 
of course be very complicated. 

It would be equivalent, accordingly, to explaining Mode of 

. avoiding 

the more simple and immediate by the more comph-sucher- 

. rors. 

cated and remote, if we were to attempt to derive sen- 
sations from the motions of masses, wholly aside from 
the consideration that the notions of mechanics are 
economical implements or expedients perfected to 
represent mechanical and not physiological or psycho- 
logical facts. If the means and aims of research were 
properly distinguished, and our expositions were re- 
stricted to the presentation of actual facts, false prob- 
lems of this kind could not arise. 

4. All physical knowledge can only mentally repre- The princi- 
sent and anticipate compounds of those elements we chancs n not 
call sensations. It is concerned with the connection of tion but 
these elements. Such an element, say the heat of a body aspect 0" 
A, is connected, not only with other elements, say with 
such whose aggregate makes up the flame B, but also 
with the aggregate of certain elements of our body, say 
with the aggregate of the elements of a nerve N. As 
simple object and element A^is not essentially, but only 
conventionally, different from A and B. The connection 
of A and B is a problem of physics, that of A and N a 
problem of physiology. Neither is alone existent ; both 
exist at once. Only provisionally can we neglect 
either. Processes, thus, that in appearance are purely 
mechanical, are, in addition to their evident mechani- 
cal features, always physiological, and, consequently, 
also electrical, chemical, and so forth. The science of 
mechanics does not comprise the foundations, no, nor 
even a part of the world, but only an aspect of it. - 

Add CONCLUSION s w y>j>. ff. V*-*! 



APPENDIX. 



(See page 3.) 



Recent research has contributed greatly to our 
knowledge of the scientific literature of antiquity, and 
our opinion of the achievements of the ancient world 
in science has been correspondingly increased. Schia- 
parelli has done much to place the work of the 
Greeks in astronomy in its right light, and Govi has 
disclosed many precious treasures in his edition of 
the Optics of Ptolemy. The view that the Greeks 
were especially neglectful of experiment can no longer 
be maintained unqualifiedly. The most ancient ex- 
periments are doubtless those of the Pythagoreans, 
who employed a monochord with moveable bridge for 
determining the lengths of strings emitting harmonic 
notes. Anaxagoras's demonstration of the corporeal- 
ity of the air by means of closed inflated tubes, and 
that of Empedocles by means of a vessel having its 
orifice inverted in water (Aristotle, Physics) are both 
primitive experiments. Ptolemy instituted systematic 
experiments on the refraction of light, while his ob- 
servations in physiological optics are still full of in- 
terest to-day. Aristotle (Meteorology) describes phe- 
nomena that go to explain the rainbow. The absurd 
stories which tend to arouse our mistrust, like that of 
Pythagoras and the anvil which emitted harmonic 



510 THE SCIENCE OF MECHANICS. 

notes when struck by hammers of different weights, 
probably sprang from the fanciful brains of ignorant 
reporters. Pliny abounds in such vagaries. But they 
are not, as a matter of fact, a whit more incorrect or 
nonsensical than the stories of Newton's falling apple 
and of Watts's tea-kettle. The situation is, more- 
over, rendered quite intelligible when we consider the 
difficulties and the expense attending the production 
of ancient books and their consequent limited circula- 
tion. The conditions here involved are concisely dis- 
cussed by J. Mueller in his paper, "Ueber das Ex- 
periment in den physikalischen Studien der Grie- 
chen," Naturwiss. Verein zu Innsbruck, XXIII., 1896- 
1897. 



(See page 8.) 

Researches in mechanics were not begun by the 
Greeks until a late date, and in no wise keep pace 
with the rapid advancement of the race in the domain 
of mathematics, and notably in geometry. Reports 
of mechanical inventions, so far as they relate to the 
early inquirers, are extremely meager. Archytas, a 
distinguished citizen of Tarentum {circa 400 B. C), 
famed as a geometer and for his employment with the 
problem of the duplication of the cube, devised me- 
chanical instruments for the description of various 
curves. As an astronomer he taught that the earth 
was spherical and that it rotated upon its axis once a 
day. As a mechanician he founded the theory of pul- 
leys. He is also said to have applied geometry to 
mechanics in a treatise on this latter science, but all 
information as to details is lacking. We are told, 
though, by Aulus Gellius (X. 12) that Archytas con- 



APPEXDIX. 511 

structed an automaton consisting of a flying dove of 
wood and presumably operated by compressed air. 
which created a great sensation. It is, in fact, char- 
acteristic of the early history of mechanics that atten- 
tion should have been first directed to its practical 
advantages and to the construction of automata de- 
signed to excite wonder in ignorant people. 

Even in the days of Ctesibius (285-247 B. C.) and 
Hero (first century A. D.) the situation had not ma- 
terially changed. So, too, during the decadence of 
civilisation in the Middle Ages, the same tendency as- 
serts itself. The artificial automata and clocks of this 
period, the construction of which popular fancy as- 
cribed to the machinations of the Devil, are well 
known. It was hoped, by imitating life outwardly, to 
ipprehend it from its inward side also. In intimate 
:onnexion with the resultant misconception of life 
stands also the singular belief in the possibility of a 
>erpetual motion. Only gradually and slowly, and in 
indistinct forms, did the genuine problems of mechan- 
ics loom up before the minds of inquirers. Aristotle's 
ract, Mechanical Problems (German trans, by Poselger, 
[annover, 1881) is characteristic in this regard. Aris- 
totle is quite adept in detecting and in formulating 
problems ; he perceived the principle of the parallel- 
ogram of motions, and was on the verge of discover- 
ing centrifugal force ; but in the actual solution of 
problems he was infelicitous. The entire tract par- 
takes more of the character of a dialectic than of a 
scientific treatise, and rests content with enunciating 
the "apories," or contradictions, involved in the prob- 
lems. But the tract upon the whole very well illus- 
trates the intellectual situation that is characteristic 
of the beginnings of scientific investigation. 



512 THE SCIENCE OF MECHANICS. 

"If a thing take place whereof the cause be not 
apparent, even though it be in accordance with na- 
ture, it appears wonderful. . . . Such are the instances 
in which small things overcome great things, small 
weights heavy weights, and incidentally all the prob- 
lems that go by the name of 'mechanical.' . . . To 
the apories (contradictions) of this character belong 
those that appertain to the lever. For it appears con- 
trary to reason that a large weight should be set in 
motion by a small force, particularly when that weight 
is in addition combined with a larger weight. A weight 
that cannot be moved without the aid of a lever can be 
moved easily with that of a lever added. The pri- 
mordial cause of all this is inherent in the nature of 
the circle, which is as one should naturally expect : 
for it is not contrary to reason that something won- 
derful should proceed out of something else that is 
wonderful. The combination of contradictory prop- 
erties, however, into a single unitary product is the 
most wonderful of all things. Now, the circle is ac- 
tually composed of just such contradictory properties. 
For it is generated by a thing that is in motion and 
by a thing that is stationary at a fixed point." 

In a subsequent passage of the same treatise there 
is a very dim presentiment of the principle of virtual 
velocities. 

Considerations of the kind here adduced give evi- 
dence of a capacity for detecting and enunciating prob- 
lems, but are far from conducting the investigator to 

their solution. 

in. 

(See page 14.) 

It may be remarked in further substantiation of 
the criticisms advanced at pages 13-14, that it is very 



APPENDIX. 513 

obvious that if the arrangement is absolutely sym- 
metrical in every respect, equilibrium obtains on the 
assumption of any form of dependence whatever of 
the disturbing factor on Z, or, generally, on the as- 
sumption P.f(L); and that consequently the particular 
form of dependence PL cannot possibly be inferred 
from the equilibrium. The fallacy of the deduction 
must accordingly be sought in the transformation to 
which the arrangement is subjected. Archimedes 
makes the action of two equal weights to be the same 
under all circumstances as that of the combined 
weights acting at the middle point of their line of 
junction. But, seeing that he both knows and as- 
sumes that distance from the fulcrum is determina- 
tive, this procedure is by the premises unpermissible, 
if the two weights are situated at unequal distances 
from the fulcrum. If a weight situated at a distance 
from the fulcrum is divided into two equal parts, and 
these parts are moved in contrary directions symmet- 
rically to their original point of support; one of the 
equal weights will be carried as near to the fulcrum 
as the other weight is carried from it. If it is assumed 
that the action remains constant during such proce- 
dure, then the particular form of dependence of the 
moment on L is implicitly determined by what has 
been done, inasmuch as the result is only possible 
provided the form be PL, or be proportional \o L. But 
in such an event all further deduction is superfluous. 
The entire deduction contains the proposition to be 
demonstrated, by assumption if not explicitly. 



5 r 4 THE SCIENCE OF MECHANICS. 

IV. 

(See page 20.) 

Experiments are never absolutely exact, but they 
at least may lead the inquiring mind to conjecture that 
the key which will clear up the connexion of all the 
facts is contained in the exact metrical expression 
PL. On no other hypothesis are the deductions of 
Archimedes, Galileo, and the rest intelligible. The 
required transformations, extensions, and compres- 
sions of the prisms may now be carried out with per- 
fect certainty. 

A knife edge may be introduced at any point un- 
der a prism suspended from its center without dis- 
turbing the equilib- 
rium (see Fig. 236), 
and several such ar- 
rangements may be 
rigidly combined to- 

zs^ gether so as to form 

Fig. 236. apparently new cases 

of equilibrium. The 
conversion and disintegration of the case of equi- 
librium into several other cases (Galileo) is possible 
only by taking into account the value of PL. I can- 
not agree with O. Holder who upholds the correct- 
ness of the Archimedean deductions against my criti- 
cisms in his essay Denken und Ans chanting in dcr Gco- 
QQO nie/n'e, although. I'am greatly pleased with the extent 
of our agreement as to the nature of the exact sci 
ences and their foundations. It would seem as if 
Archimedes (De cequiponderantibus) regarded it as a 
general experience that two equal weights may under 
all circumstances be replaced by one equal to their 



APPESDIX. 515 

combined weight at the center (Theorem 5, Corrol- 
ary 2). In such an event, his long deduction (Theo- 
rem 6) would be necessary, for the reason sought fol- 
lows immediately (see pp. 14, 513). Archimedes's 
mode of expression is not in favor of this view. 
Nevertheless, a theorem of this kind cannot be re- 
garded as a priori evident; and the views advanced 
on pp. 14, 513 appear to me to be still uncontro- 
verted. W| X $*ff'f~ ' 

v. 

(See page 79.) 

Stevinus's procedure may be looked at from still 
another point of view. If it is a fact, for our mechan- 
ical instinct, that a heavy endless chain will not ro- 
tate, then the individual simple cases of equilibrium 
on an inclined plane which Stevinus devised and 
which are readily controlled quantitatively, may be 
regarded as so many special experiences. For it is 
not essential that the experiments should have been 
actually carried out, if the result is beyond question 
of doubt. As a matter of fact, Stevinus experiments 
in thought. Stevinus's result could actually have 
been deduced from the corresponding physical exper- 
iments, with friction reduced to a minimum. In an 
analogous manner, Archimedes's considerations with 
respect to the lever might be conceived after the 
fashion of Galileo's procedure. If the various mental 
experiments had been executed physically, the linear 
dependence of the static moment on the distance of 
the weight from the axis could be deduced with per- 
fect rigor. We shall have still many instances to ad- 
duce, among the foremost inquirers in the domain of 
mechanics, of this tentative adaptation of special 



5 i6 THE SCIENCE OF MECHANICS. 

quantitative conceptions to general instinctive im- 
pressions. The same phenomena are presented in 
other domains also. I may be permitted to refer in 
this connexion to the expositions which I have given 
in my Principles of Heat, page 151. It may be said 
that the most significant and most important advances 
in science have been made in this manner. The habit 
which great inquirers have of bringing their single 
conceptions into agreement with the general concep- 
tion or ideal of an entire province of phenomena, their 
constant consideration of the whole in their treatment 
of parts, may be characterised as a genuinely philo- 
sophical procedure. A truly philosophical treatment 
of any special science will always consist in bringing 
the results into relationship and harmony with the 
established knowledge of the whole. The fanciful 
extravagances of philosophy, as well as infelicitous 
and abortive special theories, will be eliminated in 
this manner. 

It will be worth while to review again the points 
of agreement and difference in the mental procedures 
of Stevinus and Archimedes. Stevinus reached the 
very general view that a mobile, heavy, endless chain 
of any form stays at rest. He is able to deduce from 
this, general view, without difficulty, special cases, 
which are quantitatively easily controlled. The case 
from which Archimedes starts, on the other hand, is 
the most special conceivable. He cannot possibly 
deduce from his special case in an unassailable man- 
ner the behavior which may be expected under more 
general conditions. If he apparently succeeds in so 
doing, the reason is that he already knows the result 
which he is seeking, whilst Stevinus, although he too 
doubtless knows, approximately at least, what he is 



APPEXDIX. 517 

in search of, nevertheless could have found it directly 
by his manner of procedure, even if he had not known 
it. When the static relationship is rediscovered in 
such a manner it has a higher value than the result of 
a metrical experiment would have, which always de- 
viates somewhat from the theoretical truth. The de- 
viation increases with the disturbing circumstances, 
as with friction, and decreases with the diminution of 
these difficulties. The exact static relationship is 
reached by idealisation and disregard of these dis- 
turbing elements. It appears in the Archimedean and 
Stevinian procedures as an hypothesis without which 
the individual facts of experience would at once be- 
come involved in logical contradictions. Not until 
we have possessed this hypothesis can we by operat- 
ing with the exact concepts reconstruct the facts and 
acquire a scientific and logical mastery of them. The 
lever and the inclined plane are self-created ideal ob- 
jects of mechanics. These objects alone completely 
satisfy the logical demands which we make of them ; 
the physical lever satisfies these conditions only in 
measure in which it approaches the ideal lever. The 
natural inquirer strives to adapt his ideals to reality. 

VI. 

(See page no ) 

Our modern notions with regard to the nature of 
ail are a direct continuation of the ancient ideas. An- 
axagoras proves the corporeality of air from its resist- 
ance to compression in closed bags of skin, and from 
the gathering up of the expelled air (in the form of 
bubbles?) by water (Aristotle, Physics, IV., rf). Ac- 
cording to Empedocles, the air prevents the water 



A 



5 i8 THE SCIENCE OF MECHANICS. 

from penetrating into the interior of a vessel immersed 
with its aperture downwards (Gomperz, Griechische 
Denker, I., p. 191). Philo of Byzantium employs for 
the same purpose an inverted vessel having in its bot- 
tom an orifice closed with wax. The water will not 
penetrate into the submerged vessel until the wax 
cork is removed, wherupon the air escapes in bubbles. 
An entire series of experiments of this kind is per- 
formed, in almost the precise form customary in the 
schools to-day (P/iilonis lib. de ingeniis spiritualibus, in 
V. Rose's Anecdota grceca et latino). Hero describes 
in his Pneumatics many of the experiments of his 
predecessors, with additions of his own ; in theory he 
is an adherent of Strato, who occupied an intermedi- 
ate position between Aristotle and Democritus. An 
absolute and continuous vacuum, he says, can be 
produced only artificially, although numberless tiny 
vacua exist between the particles of bodies, including 
air, just as air does among grains of sand. This is 
proved, in quite the same ingenuous fashion as in our 
present elementary books, from the possibility of rare- 
fying and compressing bodies, including air (inrush- 
ing and outrushing of the air in Hero's ball). An ar- 
gument of Hero's for the existence of vacua (pores) 
between corporeal particles rests on the fact that rays 
of light penetrate water. The result of artificially in- 
creasing a vacuum, according to Hero and his prede- 
cessors, is always the attraction and solicitation of 
adjacent bodies. A light vessel with a narrow aper- 
ture remains hanging to the lips after the air has been 
exhausted. The orifice may be closed with the finger 
and the vessel submerged in water. "If the finger 
be released, the water will rise in the vacuum created, 
although the movement of the liquid upward is not 



APPENDIX. 519 

according to nature. The phenomenon of the cup- 
ping-glass is the same ; these glasses, when placed 
on the body, not only do not fall off, although they 
are heavy enough, but they also draw out adjacent 
particles through the pores of the body." The bent 
siphon is also treated at length. "The filling of the 
siphon on exhaustion of the air is accomplished by 
the liquid's closely following the exhausted air, for 
the reason that a continuous vacuum is inconceiv- 
able." If the two arms of the siphon are of the same 
length, nothing flows out. "The water is held in 
equilibrium as in a balance." Hero accordingly con- 
ceives of the flow of water as analogous to the move- 
ment of a chain hanging with unequal lengths over a 
pulley. The union of the two columns, which for us 
is preserved by the pressure of the atmosphere, is 
cared for in his case by the "inconceivability of a 
continuous vacuum." It is shown at length, not that 
the smaller mass of water is attracted and drawn 
along by the greater mass, and that conformably to 
this principle water cannot flow upwards, but rather 
that the phenomenon is in harmony with the principle 
of communicating vessels. The many pretty and in- 
genious tricks which Hero describes in his Pneumatics 
and in his Automata, and which were designed partly 
to entertain and partly to excite wonder, offer a 
charming picture of the material civilisation of the day 
rather than excite our scientific interest. The auto- 
matic sounding of trumpets and the opening of tem- 
ple doors, with the thunder simultaneously produced, 
are not matters which interest science properly so 
called. Yet Hero's writings and notions contributed 
much toward the diffusion of physical knowledge 
f compare W. Schmidt, HirJs Werkc, Leipsic, 1899, 



5 20 THE SCIENCE OF MECHANICS. 

and Diels, System des Strato, Sitzungsberichte der Ber- 
liner Akademie, 1893). 

VII. 

(See page 129.) 

It has often been asserted that Galileo had prede- 
cessors of great prominence in his method of think- 
ing, and while it is far from our purpose to gainsay 
this, we have still to emphasise the fact that Galileo 
overtowered them all. The greatest predecessor of 
Galileo, to whom we have already referred in another 
place, was Leonardo da Vinci, 1452-1519; now, it 
was impossible for Leonardo's achievements to have 
influenced the development of science at the time, for 
the reason that they were not made known in their 
entirety until the publication of Venturi in 1797. Leo- 
nardo knew the ratio of the times of descent down 
the slope and the height of an inclined plane. Fre- 
quently also a knowledge of the law of inertia is at- 
tributed to him. Indeed, some sort of instinctive 
knowledge of the persistence of motion once begun 
will not be gainsaid to any normal man. But Leo- 
nardo seems to have gone much farther than this. 
He knows that from a column of checkers one of the 
pieces may be knocked out without disturbing the 
others ; he knows that a body in motion will move 
longer according as the resistance is less, but he be- 
lieves that the body will move a distance proportional 
to the impulse, and nowhere expressly speaks of the 
persistence of the motion when the resistance is alto- 
gether removed. (Compare Wohlwill, Bibliotheca Ma- 
thematica, Stockholm, 1888, p. 19). Benedetti (1530- 
1590) knows that falling bodies are accelerated, and 
explains the acceleration as due to the summation 



of the impulses of gravity' {Divers, speculat. math, et 
physic, liber, Taurini, 1585). He ascribes the progres- 
sive motion of a projectile, not as the Peripatetics 
did, to the agency of the medium, but to the virtus 
impressa, though without attaining perfect clearness 
with regard to these problems. Galileo seems actu- 
ally to have proceeded from Benedetti's point of view, 
for his youthful productions are allied to those of 
Benedetti. Galileo also assumes a virtus impressa, 
which he conceives to decrease in efficiency, and ac- 
cording to Wohlwill it appears that it was not until 
1604 that he came into full possession of the laws of 
falling bodies. 

G. Vailati, who has devoted much attention to Be- 
nedetti's investigations (Atti della R. Acad, di Torino, 
Vol. XXXIII., 1898), finds the chief merit of Bene- 
detti to be that he subjected the Aristotelian views to 
mathematical and critical scrutiny and correction, and 
endeavored to lay bare their inherent contradictions, 
thus preparing the way for further progress. He 
knows that the assumption of the Aristotelians, that 
the velocity of falling bodies is inversely proportional 
to the density of the surrounding medium, is un- 
tenable and possible only in special cases. Let the 
velocity of descent be proportional to / q, where/ 
is the weight of the body and q the upward impulsion 
due to the medium. If only half the velocity of de- 
scent is set up in a medium of double the density, 
the equation/ q = 1{p 2q) must exist, a relation 
which is possible only in case p = 3q. Light bodies 
per se do not exist for Benedetti; he ascribes weight 
and upward impulsion even to air. Different-sized 
bodies of the same material fall, in his opinion, with 
the same velocity. Benedetti reaches this result by 



(]S&\ 



522 THE SCIENCE OF MECHANICS 

conceiving equal bodies falling alongside each other 
first disconnected and then connected, where the con- 
nexion cannot alter the motion. In this he approaches 
to the conception of Galileo, with the exception that 
the latter takes a profounder view of the matter. 
Nevertheless, Benedetti also falls into many errors; 
he believes, for example, that the velocity of descent 
of bodies of the same size and of the same shape is 
proportional to their weight, that is, to their density. 
His reflexions on catapults, no less than his views on 
the oscillation of a body about the center of the earth 
in a canal bored through the earth, are interesting, 
and contain little to be criticised. Bodies projected 
horizontally appear to approach the earth more slowly. 
Benedetti is accordingly of the opinion that the force 
of gravity is diminished also in the case of a top rotat- 
ing with its axis in a vertical position. He thus does 
not solve the riddle fully, but prepares the way for 
the solution. 

VIII. 
(See page 134.) 

If we are to understand Galileo's train of thought, 
we must bear in mind that he was already in posses- 
sion of instinctive experiences prior to his resorting 
to experiment. 

Freely falling bodies are followed with more diffi- 
culty by the eye the longer and the farther they have 
fallen ; their impact on the hand receiving them is in 
like measure sharper ; the sound of their striking 
louder. The velocity accordingly increases with the 
time elapsed and the space traversed. But for scien- 
tific purposes our mental representations of the facts 
of sensual experience must be submitted to conceptual 



APPENDIX. 523 

formulation. Only thus may they be used for discov- 
ering by abstract mathematical rules unknown prop- 
erties conceived to be dependent on certain initial 
properties having definite and assignable arithmetic 
values; or, for completing what has been only partly 
given. This formulation is effected by isolating and 
emphasising what is deemed of importance, by neg- 
lecting what is subsidiary, by abstracting, by idealis- 
ing. The experiment determines whether the form 
chosen is adequate to the facts. Without some pre- 
conceived opinion the experiment is impossible, be- 
cause its form is determined by the opinion. For 
how and on what could we experiment if we did not 
previously have some suspicion of what we were 
about ? The complemental function which the experi- 
ment is to fulfil is determined entirely by our prior 
experience. The experiment confirms, modifies, or 
overthrows our suspicion. The modern inquirer would 
ask in a similar predicament : Of what is v a function? 
What function of / is r? Galileo asks, in his ingenu- 
ous and primitive way: is v proportional to s, is v 
proportional to t? Galileo, thus, gropes his way along 
synthetically, but reaches his goal nevertheless. Sys- 
tematic, routine methods are the final outcome of re- 
search, and do not stand perfectly developed at the 
disposal of genius in the first steps it takes. (Com- 
pare the article " Ueber Gedankenexperimente," Zeit- 
schri/t/iir den/>/ns. ttnd chem. Unterricht, 1897, I.) 

j " re f' *ff' /*? 

IX. 
(See page 140.) 

In an exhaustive study in the Zeitschrift fiir Volker- 
psychologie, 1884, Vol. XIV., pp. 365-410, and Vol. 



524 THE SCIENCE OF MECHANICS. 

XV., pp. 70-135, 337-387, entitled "Die Entdeckung 
des Beharrungsgesetzes," E. Wohlwill has shown that 
the predecessors and contemporaries of Galileo, nay, 
even Galileo himself, only very gradually abandoned 
the Aristotelian conceptions for the acceptance of the 
law of inertia. Even in Galileo's mind uniform cir- 
cular motion and uniform horizontal motion occupy 
distinct places. Wohlwill's researches are very ac- 
ceptable and show that Galileo had not attained per- 
fect clearness in his own new ideas and was liable to 
frequent reversion to the old views, as might have 
been expected. 

Indeed, from my own exposition the reader will 
have inferred that the law of inertia did not possess 
in Galileo's mind the degree of clearness and univer- 
sality that it subsequently acquired. (See pp. 140 and 
143.) With regard to my exposition at pages 140- 
141, however, I still believe, in spite of the opinions 
of Wohlwill and Poske, that I have indicated the 
point which both for Galileo and his successors must 
have placed in the most favorable light the transition 
from the old conception to the new. How much was 
wanting to absolute comprehension, may be gathered 
from the fact that Baliani was able without difficulty 
to infer from Galileo's statement that acquired velo- 
city could not be destroyed, a fact which Wohlwill 
himself points out (p. 112). It is not at all surpris- 
ing that in treating of the motion of heavy bodies, 
Galileo applies his law of inertia almost exclusively 
to horizontal movements. Yet he knows that a mus- 
ket-ball possessing no weight would continue rectiline- 
arly on its path in the direction of the barrel. {Dia- 
logues on the two World Systems, German translation, 
Leipsic, 1891, p. 184.) His hesitation in enunciating 



APPENDIX. 525 

in its most general terms a law that at first blush ap- 
pears so startling, is not surprising. 

x. 

(See page 155.) 

We cannot adequately appreciate the extent of 
Galileo's achievement in the analysis of the motion of 
projectiles until we examine his predecessors' endeav- 
ors in this field. Santbach (1561) is of opinion, that a 
cannon-ball speeds onward in a straight line until its 
velocity is exhausted and then drops to the ground in 
a vertical direction. Tartaglia (1537) compounds the 
path of a projectile out of a straight line, the arc of a 
circle, and lastly the vertical tangent to the arc. He 
is perfectly aware, as Rivius later (1582) more dis- 
tinctly states, that accurately viewed the path is 
curved at all points, since the deflective action of 
gravity never ceases; but he is yet unable to arrive 
at a complete analysis. The initial portion of the 
path is well calculated to arouse the illusive impres- 
sion that the action of gravity has been annulled by f 
the velocity of the projection, an illusion to which 
even Benedetti fell a victim. (See Appendix, vn., p. 
129.) We fail to observe any descent in the initial 
part of the curve, and forget to take into account the 
shortness of the corresponding time of the descent. 
By a similar oversight a jet of water may assume the 
appearance of a solid body suspended in the air, if 
one is unmindful of the fact that it is made up of a 
mass of rapidly alternating minute particles. The 
same illusion is met with in the centrifugal pendulum, 
in the top, in Aitken's flexible chain rendered rigid 
by rapid rotation {Philosophical Magazine, 1878), in 
the locomotive which rushes safely across a defective 



526 THE SCIENCE OE MECHANICS. 

bridge, through which it would have crashed if at 
rest, but which, owing to the insufficient time of des- 
cent and of the period in which it can do work, leaves 
the bridge intact. On thorough analysis none of 
these phenomena are more surprising than the most 
ordinary events. As Vailati remarks, the rapid spread 
of firearms in the fourteenth century gave a distinct 
impulse to the study of the motion of projectiles, and 
indirectly to that of mechanics generally. Essentially 
the same conditions occur in the case of the ancient 
catapults and in the hurling of missiles by the hand, 
but the new and imposing form of the phenomenon 
doubtless exercised a great fascination on the curios- 
ity of people. 

So much for history. And now a word as to the 
notion of "composition." Galileo's conception of the 
motion- of a projectile as a process compounded of 
two distinct and independent motions, is suggestive 
of an entire group of similar important episfcemologi- 
cal processes.^ We may say that it is as important to 
perceive the non drpendence of two circumstances A 
and B on each other, as it is to perceive the depe?idence 
of two circumstances A and C on each other. For 
the first perception alone enables us to pursue the 
second relation with composure. Think only of how- 
serious an obstacle the assumption of non-existing 
tausal relations constituted to the research of the 
Middle Ages. Similar to Galileo's discovery is that 
of the parallelogram of forces by Newton, the compo- 
sition of the vibrations of strings by Sauveur, the com 
position of thermal disturbances by Fourier. Through 
this latter inquirer the method of compounding a phe- 
nomenon out of mutually independent partial phe 
nomena by means of representing a general integral 



APPEXDIX. 527 

as the sum of particular integrals has penetrated into 
every nook and corner of physics. The decomposi- 
tion of phenomena into mutually independent parts 
has been aptly characterised by P. Volkmann as iso- 
lation, and the composition of a phenomenon out of 
such parts, superposition. The two processes combined 
enable us to comprehend, or reconstruct in thought, 
piecemeal, what, as a whole, it would be impossible 
for us to grasp. 

"Nature with its myriad phenomena assumes a 
unified aspect only in the rarest cases; in the major- 
ity of instances it exhibits a thoroughly composite 
character . . . ; it is accordingly one of the duties of 
science to conceive phenomena as made up of sets of 
partial phenomena, and at first to study these partial 
phenomena in their purity. Not until we know to 
what extent each circumstance shares in the phenom- 
enon as an entirety do we acquire a command over 
the whole. ..." (Cf. P. Volkmann, Erkenntnisstheo- 
retische Grundziige der Naturwissenschaft, 1896, p. 70. 
Cf. also my Principles of Heat, German edition, pp. 
123, 151. 452). 

XI. 
(See page 161.) 

The perspicuous deduction of the expression for 
centrifugal force based on the principle of Hamilton's 
hodograph may also be mentioned. If a body move 
uniformly in a circle of radius r (Fig. 237), the velo- 
city 7/ at the point A of the path is transformed by 
the traction of the string into the velocity v of like 
magnitude but different direction at the point B. If 
from O as centre (Fig. 238) we lay off as to magni- 
tude and direction all the velocities the body succes- 






528 



THE SCIENCE OF MECHANICS. 



sively acquires, these lines will represent the sum 
of the radii v of the circle. For OM to be trans- 
formed into ON, the perpendicular component to it, 
MN, must be added. During the period of revolu- 
tion T the velocity is uniformly increased in the direc- 
tions of the radii r by an amount 2nv. The numeri- 




M N 




Fig. 237. 



Fig. 238. 



Fig. 230. 



cal measure of the radial acceleration is therefore 

2 7T7) 7) 

<=, and since vT=2nr, therefore also <p= . 

If to OM=v the very small component w is added 
(Fig. 239), the resultant will strictly be a greater 



velocity V v' 1 -\- w 2 =z>-|- , as the approximate ex- 
traction of the square root will show. But on contin- 

to 
uous deflection ^- vanishes with respect to v ; hence, 

2v 

only the direction, but not the magnitude, of the 
velocity changes. 



XII. 
(See page 162.) 

Even Descartes thought of explaining the centri- 
petal impulsion of floating bodies in a vortical me- 
dium, after this manner. But Huygens correctly re- 



APPENDIX. 529 

marked that on this hypothesis we should have to 
assume that the lightest bodies received the greatest 
centripetal impulsion, and that all heavy bodies would 
without exception have to be lighter than the vortical 
medium. Huygens observes further that like phe- 
nomena are also necessarily presented in the case of 
bodies, be they what they ma) r , that do not participate 
in the whirling movement, that is to say, such as 
might exist without centrifugal force in a vortical 
medium affected with centrifugal force. For exam- 
ple, a sphere composed of any material whatsoever 
but moveable only along a stationary axis, say a wire, 
is impelled toward the axis of rotation in a whirling 
medium. 

In a closed vessel containing water Huygens 
placed small particles of sealing wax which are 
slightly heavier than water and hence touch the bot- 
tom of the vessel. If the vessel be rotated, the par- 
ticles of sealing wax will flock toward the outer rim 
of the vessel. If the vessel be then suddenly brought 
to rest, the water will continue to rotate while the 
particles of sealing wax which touch the bottom and 
are therefore more rapidly arrested in their move- 
ment, will now be impelled toward the axis of the 
vessel. In this process Huygens saw an exact replica 
of gravity. An ether whirling in one direction only, 
did not appear to fulfil his requirements. Ultimately, 
he thought, it would sweep everything with it. He 
accordingly assumed ether-particles that sped rapidly 
about in all directions, it being his theory that in a 
closed space, circular, as contrasted with radial, mo- 
tions would of themselves prepondarate. This ether 
appeared to him adequate to explain gravity. The 
detailed exposition of this kinetic theory of gravity is 



530 THE SCIENCE OF MECHANICS. 

found in Huygens's tract On the Cause of Gravitation 
(German trans, by Mewes, Berlin, 1893). See also 
Lasswitz, Geschichte der Atomistik, 1890, Vol. II., p. 
344- 

XIII. 

(See page 187 ) 

It has been impossible for us to enter upon the 
signal achievements of Huygens in physics proper. 
But a few points may be briefly indicated. He is the 
creator of the wave-theory of light, which ultimately 
overthrew the emission theory of Newton. His at- 
tention was drawn, in fact, to precisely those features 
of luminous phenomena that had escaped Newton. 
With respect to physics he took up with great enthu- 
siasm the idea of Descartes that all things were to be 
explained mechanically, though without being blind 
to its errors, which he acutely and correctly criticised. 
His predilection for mechanical explanations renderec 
him also an opponent of Newton's action at a distance, 
which he wished to replace by pressures and impacts, 
that is, by action due to contact. In his endeavor tc 
do so he lighted upon some peculiar conceptions, like 
that of magnetic currents, which at first could not 
compete with the influential theory of Newton, but 
has recently been reinstated in its full rights in tht 
unbiassed efforts of Faraday and Maxwell. As 
geometer and mathematician also Huygens is to be 
ranked high, and in this connexion reference need be 
made only to his theory of games of chance. His 
astronomical observations, his achievements in theo- 
retical and practical dioptrics advanced these depart- 
ments very considerably. As a technicist he is the 
inventor of the powder-machine, the idea of which 



APPEXDIX. 531 

has found actualisation in the modern gas-machine. 
As a physiologist he surmised the accommodation of 
the eye by deformation of the lens. All these things 
can scarcely be mentioned here. Our opinion of Huy- 
gens grows as his labors are made better known by the 
complete edition of his works. A brief and reveren- 
tial sketch of his scientific career in all its phases is 
given by J. Bosscha in a pamphlet entitled Christian 
Huyghens, Rede am 200. Geddchtnisstage seines Lebens- 
endes, German trans, by Engelmann, Leipsic, 1895. 

XIV. 
(See page 190.) 

Rosenberger is correct in his statement {Newton 
und seine physikalischen Principien, 1895") that the idea 
of universal gravitation did not originate with New- 
ton, but that Newton had many highly deserving pred- 
ecessors. But it may be safely asserted that it was, 
with all of them, a quesij^Jk^njecture, of a groping 
and imperfect grasp o^H ^^^P^i, and that no one 
before Newton grapple^JIP^he notion so compre- 
hensively and energetically; so that above and beyond 
the great mathematical problem, which Rosenberger 
concedes, there still remains to Newton the credit of 
a colossal feat of the imagination. 

Among Newton's forerunners may first be men- 
tioned Copernicus, who (in 1543) says: "I am at 
least of opinion that gravity is nothing more than a 
natural tendency implanted in particles by the divine 
providence of the Master of the Universe, by virtue of 
which, they, collecting together in the shape of a 
sphere, do form their own proper unity and integrity. 
And it is to be assumed that this propensity is in- 
herent also in the sun, the moon, and the other plan- 



532 THE SCIENCE OF MECHANICS. 

ets." Similarly, Kepler (1609), like Gilbert before 
him (1600), conceives of gravity as the analogue of 
magnetic attraction. By this analogy, Hooke, it seems, 
is led: to the notion of a diminution of gravity with the 
distance ; and in picturing its action as due to a kind 
of radiation, he even hits upon the idea of its acting \ 
inversely as the square of the distance. He even 
sought to determine the diminution of its effect (1686) 
b)' weighing bodies hung at different heights from the 
top of Westminster Abbey (precisely after the more 
modern method of Jolly), by means of spring- balances 
and pendulum clocks, but of course without results. 
The conical pendulum appeared to him admirably 
adapted for illustrating the motion of the planets. 
Thus Hooke really approached nearest to Newton's 
conception, though he never completely reached the 
latter's altitude of view. 

In two instructive writings {Kepler's Lehre von dcr 
Gravitation, Halle, ijrijftf f ^Gravitation bci Galileo 
u. Borelli, Berlin, i8^J '.: -^^>eck investigates the 

early history of the docti^BP^ravitation with Kepler 
on the one hand and Galileo and Borelli on the other. 
Despite his adherence to scholastic, Aristotelian no- 
tions, Kepler has sufficient insight to see that there is 
a real physical problem presented by the phenomena 
of the planetary system; the moon, in his view, is 
swept along with the earth in its motion round the 
sun, and in its turn drags the tidal wave along with 
it, just as the earth attracts heavy bodies. Also, for 
the planets the source of motion is sought in the sun, 
from which immaterial levers extend that rotate with 
the sun and carry the distant planets around more 
slowly than the near ones. By this view, Kepler was 
enabled to guess that the period of rotation of the sun 



\ 



APPENDIX. 533 

was less than eighty-eight days, the period of revolu- 
tion of Mercury. At times, the sun is also conceived 
as a revolving magnet, over against which are placed 
the magnetic planets. In Galileo's conception of the 
universe, the formal, mathematical, and esthetical 
'point of view predominates. He rejects each and 
every assumption of attraction, and even scouted the 
ilea as childish in Kepler. The planetary system had 
not yet taken the shape of a genuine physical problem 
for him. Yet he assumed with Gilbert that an imma- 
terial geometric point can exercise no physical action, 
and he did very much toward demonstrating the ter- 
restrial nature of the heavenly bodies. Borelli (in his 
work on the satellites of the Jupiter) conceives the 
planets as floating between layers of ether of differing 
densities. They have a natural tendency to approach 
their central body, (the term attraction is avoided,) 
which is offset by the centrifugal force set up by the 
revolution. Borelli illustrates his theory by an experi- 
ment very similar to that described by us in Fig. 106, 
p. 162. As will be seen, he approaches very closely 
to Newton. His theory is, though, a combination of 
Descartes's and Newton's. 

xv. 

(See page 191.) 

Newton illustrated the identity of terrestrial grav- 
ity with the universal gravitation that determined the 
motions of the celestial bodies, as follows. He con- 
ceived a stone to be hurled with successive increases 
of horizontal velocity fjrom the top of a high moun- 
tain. Neglecting the resistance of the air, the para- 
bolas successively described by the stone will increase 
in length until finally tbey will fall clear of the earth- 



534 THE SCIENCE OF MECHANICS. 

altogether, and the stone will be converted into a 
satellite circling round the earth. Newton begins with 
the fact of universal gravity. An explanation of the 
phenomenon was not forthcoming, and it was not his 
wont, he says, to frame hypotheses. Nevertheless he 
could not set his thoughts at rest so easily, as is ap- 
parent from his well-known letter to Bentley. That 
gravity was immanent and innate in matter, so that 
one body could act on another directly through empty 
space, appeared to him absurd. But he is unable to 
decide whether the intermediary agency is material or 
immaterial (spiritual?). Like all his predecessors and 
successors, Newton felt the need of explaining gravi- 
tation, by some such means as actions of contact. Yet 
the great success which Newton achieved in astron- 
omy with forces acting at a distance as the basis of 
deduction, soon changed the situation very consider- 
ably. Inquirers accustomed themselves to these forces 
as points of departure for their explanations and the 
impulse to inquire after their origin soon disappeared 
almost completely. The attempt was now made to 
introduce these forces into all the departments of 
physics, by conceiving bodies to be composed of par- 
ticles separated by vacuous interstices and thus acting 
on one another at a distance. Finally even, the re- 
sistance of bodies to pressure and impact, this is to 
say, even forces of contact, were explained by forces 
acting at a distance between particles. As a fact, the 
functions representing the former are more compli- 
cated than those representing the latter. 

The doctrine of forces acting at a distance doubt- 
less stood in highest esteem with Laplace and his 
contemporaries. Faraday's unbiassed and ingenious 
conceptions and Maxwell's mathematical formulation 



APPENDIX. 535 

of them again turned the tide in favor of the forces 
of contact. Divers difficulties had raised doubts' in 
the minds of astronomers as to the exactitude of New- 
ton's law, and slight quantitative variations of it were 
looked for. After it had been demonstrated, however, 
that electricity travelled with finite velocity, the ques- 
tion of a like state of affairs in connexion with the 
analogous action of gravitation again naturally arose. 
As a fact, gravitation bears a close resemblance to 
electrical forces acting at a distance, save in the single 
respect that so far as we know, attraction only and 
not repulsion takes place in the case of gravitation. 
Foppl ("Ueber eine Erweiterung des Gravitations- 
gesetzes," Sitzungsber. d. Munch. Akad., 1897, p. 6 et 
seq.) is of opinion, that we may, without becoming 
involved in contradictions, assume also with respect 
to gravitation negative masses, which attract one an- 
other but repel positive masses, and assume therefore 
z\so finite fields of gravitation, similar to the electric 
fields. Drude (in his report on actions at a distance 
made for the German Naturforscherversammlung of 
1897) enumerates many experiments for establishing 
a velocity of propagation for gravitation, which go 
back as far as Laplace. The result is to be regarded 
is a negative one, for the velocities which it is at all 
>ossible to consider as such, do not accord with one 
another, though they are all very large multiples of 
the velocity of light. Paul Gerber alone (" Ueber die 
raumliche u. zeitliche Ausbreitung der Gravitation," 
Zeitschrift f. Math. u. Phys., 1898, II.), from the peri- 
helial motion of Mercury, forty-one seconds in a cen- 
tury, finds the velocity of propagation of gravitation 
to be the same as that of light. This would speak in 
favor of the ether as the medium of gravitation. (Com- 



536 THE SCIENCE OE MECHANICS. 

pare W. Wien, "Ueber die Moglichkeit einer elektro- 
magnetischen Begriindung der Mechanik," Archives 
Ne'er landaises, The Hague, 1900, V., p. 96.) 

XVI. 
(See page 195.) 

It should be observed that the notion of mass as 
quantity of matter was psychologically a very natural 
conception for Newton, with his peculiar develop- 
ment. Critical inquiries as to the origin of the con- 
cept of matter could not possibly be expected of a 
scientist in Newton's day. The concept developed 
quite instinctively ; it is discovered as a datum per- 
fectly complete, and is adopted with absolute ingenu- 
ousness. The same is the case with the concept of 
force. But force appears conjoined with matter. And, 
inasmuch as Newton invested all material particles 
with precisely identical gravitational forces, inasmuch 
as he regarded the forces exerted by the heavenly 
bodies on one another as the sum of the forces of the 
individual particles composing them, naturally these 
forces appear to be inseparably conjoined with the 
quantity of matter. Rosenberger has called attention 
to this fact in his book, Newton und seine physikalischen 
Principien (Leipzig, 1895, especially page 192). 

I have endeavored to show elsewhere {Analysis of 
the Sensations, Chicago, 1897) how starting from the 
constancy of the connexion between different sensa- 
tions we have been led to the assumption of an abso- 
lute constancy, which we call substance, the most ob- 
vious and prominent example being that of a moveable 
body distinguishable from its environment. And see- 
ing that such bodies are divisible into homogeneous 
parts, of which each presents a constant complexus 



APPENDIX. 537 

of properties, we are induced to form the notion of a 
substantial something that is quantitatively variable, 
which we call matter. But that which we take away 
from one body, makes its appearance again at some 
other place. The quantity of matter in its entirety, 
thus, proves to be constant. Strictly viewed, how- 
ever, we are concerned with precisely as many sub- 
stantial quantities as bodies have properties, and 
there is no other function left for matter save that of 
representing the constancy of connexion of the several 
properties of bodies, of which man is one only. (Com- 
pare my Principles of Heat, German edition, 1896, 
page 425.) 

xvn. 

(See page 216. 

Of the theories of the tides enunciated before 
Newton, that of Galileo alone may be briefly men- 
tioned. Galileo explains the tides as due to the rela- 
tive motion of the solid and liquid parts of the earth, 
and regards this fact as direct evidence of the motion 
of the earth and as a cardinal 
argument in favor of the Co- 
pernican system. If the earth 
(Fig. 240) rotates from the 
west to the east, and is affected 
at the same time with a pro- 
gressional motion, the parts of 
the earth at a will move with the sum, and the parts 
at b with the difference, of the two velocities. The 
water in the bed of the ocean, which is unable to fol- 
low this change in velocity quickly enough, behaves 
like the water in a plate swung rapidly back and forth, 
or like that in the bottom of a skiff which is rowed 




538 THE SCIENCE OF MECHANICS. 

with rapid alterations of speed : it piles up now in 
the front and now at the back. This is substantially 
the view that Galileo set forth in the Dialogue on the 
Two World Systems. Kepler's view, which supposes 
attraction by the moon, appears to him mystical and 
childish. He is of the opinion that it should be rele- 
gated to the category of explanations by "sympathy" 
and "antipathy," and that it admits as easily of refu- 
tation as the doctrine according to which the tides 
are created by radiation and the consequent expansion 
of the water. That on his theory the tides rise only 
once a day, did not, of course, escape Galileo's atten- 
tion. But he deceived himself with regard to the 
difficulties involved, believing himself able to explain 
the daily, monthly, and yearly periods by considering 
the natural oscillations of the water and the altera- 
tions to which its motions are subject. The principle 
of relative motion is a correct feature of this theory, 
but it is so infelicitously applied that only an ex- 
tremely illusive theory could result. We will first 
convince ourselves that the conditions supposed to 
be involved would not have the effect ascribed to 
them. Conceive a homogeneous sphere of water; 
any other effect due to 'rotation than that of a corres- 
ponding oblateness we should not expect. Now, sup- 
pose the ball to acquire in addition a uniform motion 
of progression. Its various parts will now as before 
remain at relative rest with respect to one another. 
For the case in question does not differ, according to 
our view, in any essential respect from the preceding, 
inasmuch as the progressive motion of the sphere 
may be conceived to be replaced by a motion in the 
opposite direction of all surrounding bodies. Even 
for the person who is inclined to regard the motion 



APPENDIX. 539 

as an "absolute" motion, no change is produced in 
the relation of the parts to one another by uniform 
motion of progression. Now, let us cause the sphere, 
the parts of which have no tendency to move with re- 
spect to one another, to congeal at certain points, so 
that sea-beds with liquid water in them are produced. 
The undisturbed uniform rotation will continue, and 
consequently Galileo's theory is erroneous. 

But Galileo's idea appears at first blush to be ex- 
tremely plausible; how is the paradox explained? It 
is due entirely to a negative conception of the law of 
inertia. If we ask what acceleration the water expe- 
riences, everything is clear. Water having no weight 
would be hurled off at the beginning of rotation ; 
water having weight, on the other hand, would de 
scribe a central motion around the center of the earth 
With its slight velocity of rotation it would be forced 
more and more toward the center of the earth, with 
just enough of its centripetal acceleration counter- 
acted by the resistance of the mass lying beneath, 
as to make the remainder, conjointly with the given 
tangential velocity, sufficient for motion in a circle. 
Looking at it from this point of view, all doubt and 
obscurity vanishes. But it must in justice be added 
that it was almost impossible for Galileo, unless his 
genius were supernatural, to have gone to the bottom 
of the matter. He would have been obliged to antici- 
pate the great intellectual achievements of Huygens 
and Newton, fei i<) *||| f Suf*. V>. 3| 

XVIII. 
(See page 218.) 

H. Streintz's objection {Die physikalischen Grund- 
lagen der Mechanik, Leipsic, 1883, p. 117), that a com- 



540 THE SCIENCE OF MECHANICS. 

parison of masses satisfying my definition can be ef- 
fected only by astronomical means, I am unable to 
admit. The expositions on pages 202, 218-221 amply 
refute this. Masses produce in each other accelera- 
tions in impact, as well as when subject to electric 
and magnetic forces, and when connected by a string 
on Atwood's machine. In my Elements of Physics 
(second German edition, 1891, page 27) I have shown 
how mass-ratios can be experimentally determined on 
a centrifugal machine, in a very elementary and pop- 
ular manner. The criticism in question, therefore, 
may be regarded as refuted. 

My definition is the outcome of an endeavor to 
establish the interdependence of phenomena and to re- 
move all metaphysical obscurity, without accomplish- 
ing on this account less than other definitions have 
done. I have pursued exactly the same course with 
respect to the ideas, " quantity of electricity " ("On 
the Fundamental Concepts of Electrostatics," 1883, 
Popular Scientific Lectures, Open Court Pub. Co., Chi- 
sago, 1898), "temperature," "quantity of heat" (Zcit- 
schrift fiir den physikalischen und chemischen Unterricht, 
Berlin, 1888, No. 1), and so forth. With the view 
here taken of the concept of mass is associated, how- 
ever, another difficulty, which must also be carefully 
noted, if we would be rigorously critical in our analy- 
sis of other concepts of physics, as for example the 
concepts of the theory of heat. Maxwell made refer- 
ence to this point in his investigations of the concept 
of temperature, about the same time as I did with re- 
spect to the concept of heat. I would refer here to 
the discussions on this subject in my Principles of 
Heat (German edition, Leipsic, 1896), particularly 
page 41 and page 190. 



APPESDIX. 541 

XIX. 

(See page 226.; 

My views concerning physiological time, the sen- 
sation of time, and partly also concerning physical 
time, I have expressed elsewhere (see Analysis of the 
Sensations, 1886, Chicago, Open Court Pub. Co., 1897, 
pp. 109-118, 179-181). As in the study of thermal 
phenomena we take as our measure of temperature an 
arbitrarily chosen indicator of volume, which varies in 
almost parallel correspondence with our sensation of 
heat, and which is not liable to the uncontrollable 
disturbances of our organs of sensation, so, for simi- 
lar reasons, we select as our measure of time an arbi- 
trarily chosen motion, (the angle of the earth's rotation, 
or path of a free body,) which proceeds in almost 
parallel correspondence with our sensation of time. 
If we have once made clear to ourselves that we are 
concerned only with the ascertainment of the inter- 
dependence of phenomena, as I pointed out as early as 
1 865 ( Ucbcr den Zeitsinn des Ohres, Siizungsbcrichte der 
Wiener Akademie) and 1866 (Fichte's Zeitschrift fur 
Philosophic), all metaphysical obscurities disappear. 
(Compare J Epstein, Die logischen Principien der Zeit- 
niessung, Berlin, 1887.) 

I have endeavored also {Principles of Heat, German 
edition, page 51) to point out the reason for the natu- 
ral tendency of man to hypostatise the concepts which 
have great value for him, particularly those at which 
he arrives instinctively, without a knowledge of their 
development. The considerations which I there ad- 
duced for the concept of temperature may be easily 
applied to the concept of time, and render the origin 



542 THE SCIENCE OF MECHANICS. 

of Newton's concept of "absolute" time intelligible. 
Mention is also made there (page 338) of the connex- 
ion obtaining between the concept of energy and the 
irreversibility of time, and the view is advanced that 
the entropy of the universe, if it could ever possibly 
be determined, would actually represent a species of 
absolute measure of time. I have finally to refer here 
also to the discussions of Petzoldt ("Das Gesetz der 
Eindeutigkeit, " Vierteljahrsschrijt fur wissenschaftliche 
Philosophic, 1894, page 146), {j:o which I shall reply in 
another place.) S^^^dtdtl ^ ' C ^ SL ^ ~ SujJo. ( 

vtJ + vitc.X*! ~ XX - 

-/ ... (See page 238.) 

Of the treatises which have appeared since 1883 
on the law of inertia, all of which furnish welcome 
evidence of a heightened interest in this question, I 
can here only briefly mention that of Streintz {Physi- 
kalische Grundlagen der Mechanik, Leipsic, 1883) and 
that of L. Lange {Die geschichtliche Entwicklung des 
Bewegungsbegriffes, Leipsic, 1886). 

The expression "absolute motion of translation" 
Streintz correctly pronounces as devoid of meaning 
and consequently declares certain analytical deduc- 
tions, to which he refers, superfluous. On the other 
hand, with respect to rotation, Streintz accepts New- 
ton's position, that absolute rotation can be distin- 
guished from relative rotation. In this point of view, 
therefore, one can select every body not affected with 
absolute rotation as a body of reference for the ex- 
pression of the law of inertia. 

I cannot share this view. For me, only relative 
motions exist {Erhaltung der Arbeit, p. 48; Science of 
Mechanics, p. 229), and I can see, in this regard, no 



APPENDIX. 543 

distinction between rotation and translation. When a 
body moves relatively to the fixed stars, centrifugal 
forces are produced ; when it moves relatively to some 
different body, and not relatively to the fixed stars, 
no centrifugal forces are produced. I have no objec- 
tion to calling the first rotation "absolute" rotation, 
if it be remembered that nothing is meant by such a 
designation except relative rotation with respect to the 
fixed stars. Can we fix Newton's bucket of water, 
rotate the fixed stars, and then prove the absence of 
centrifugal forces? 

The experiment is impossible, the idea is mean- 
ingless, for the two cases are not, in sense-perception, 
distinguishable from each other. I accordingly re- 
gard these two cases as the same case and Newton's 
distinction as an illusion {Science of Mechanics, page 
232)- 

But the statement is correct that it is possible to 
find one's bearings in a balloon shrouded in fog, by 
means of a body which does not rotate with respect 
to the fixed stars. But this is nothing more than an 
indirect orientation with respect to the fixed stars ; it 
is a mechanical, substituted for an optical, orienta- 
tion. 

I wish to add the following remarks in answer to 
Streintz's criticism of my view. My opinion is not to 
be confounded with that of Euler (Streintz, pp. 7, 50), 
who, as Lange has clearly shown, never arrived at 
any settled and intelligible opinion on the subject. 
Again, I never assumed that remote masses only, and 
not near ones, determine the velocity of a body 
(Streintz, p. 7); I simply spoke of an influence inde- 
pendent of distance. In the light of my expositions 
at pages 222-245, tne unprejudiced and careful reader 



544 THE SCIENCE OF MECHANICS. 

will scarcely maintain with Streintz (p. 50), that after 
so long a period of time, without a knowledge of 
Newton and Euler, I have only been led to views 
which these inquirers so lorg ago held, but were 
afterwards, partly by them and partly by others, re- 
jected. Even my remarks of 1872, which were all 
that Streintz knew, cannot justify this criticism. These 
were, for good reasons, concisely stated, but they are 
by no means so meagre as they must appear to one 
who knows them only from Streintz's criticism. The 
point of view which Streintz occupies, I at that time 
expressly rejected. 

Lange's treatise is, in my opinion, one of the best 
that have been written on this subject. Its methodi- 
cal movement wins at once the reader's sympathy. Its 
careful analysis and study, from historical and criti- 
cal points of view, of the concept of motion, have 
produced, it seems to me, results of permanent value. 
I also regard its clear emphasis and apt designation 
of the principle of "particular determination" as a 
point of much merit, although the principle itself, as 
well as its application, is not new. The principle is 
really at the basis of all measurement. The choice of 
the unit of measurement is convention ; the number 
of measurement is a result of inquiry. Every natural 
inquirer who is clearly conscious that his business is 
simply the investigation of the interdependence of 
phenomena, as I formulated the point at issue a long 
time ago (1 865-1866), employs this principle. When, 
for example {Mechanics, p. 218 et seq.), the negative 
inverse ratio of the mutually induced accelerations of 
two bodies is called the mass-ratio of these bodies, 
this is a convention, expressly acknowledged as arbi- 
trary ; but that these ratios are independent of the 



APPENDIX. 545 

kind and of the order of combination of the bodies is 
a result of inquiry. I might adduce numerous similar 
nstances from the theories of heat and electricity as 
well as from other provinces. Compare Appendix II. 

Taking it in its simplest and most perspicuous 
form, the law of inertia, in Lange's view, would read 
as follows : 

Three material points, P\, P it P z , are simultane- 
ously hurled from the same point in space and then 
left to themselves. The moment we are certain that 
the points are not situated in the same straight line, 
we join each separately with any fourth point in space, 
Q. These lines of junction, which we may respec- 
tively call G\, G? t Gz, form, at their point of meeting, 
a three-faced solid angle. If now we make this solid 
angle preserve, with unaltered rigidity, its form, and 
constantly determine in such a manner its position, 
that Pi shall always move on the line G\, P? on the 
line G-2, Pz on the line c7 3 , these lines may be regarded 
as the axis of a coordinate or inertial system, with 
respect to which every other material point, left to it- 
self, will move in a straight line. The spaces de- 
scribed by the free points in the paths so determined 
will be proportional to one another. 

A system of coordinates with respect to which 
three material points move in a straight line is, ac- 
cording to Lange, under the assumed limitations, a 
simple convention. That with respect to such a system 
also a fourth or other free material point will move in 
a straight line, and that the paths of the different 
points will all be proportional to one another, are re- 
sults of inquiry. 

In the first place, we shall not dispute the fact 
that the law of inertia can be referred to such a system 



546 THE SCIENCE OF MECHANICS. 

of time and space coordinates and expressed in this 
form. Such an expression is less fit than Streintz's 
for practical purposes, but, on the other hand, is, for 
its methodical advantages, more attractive. It espe- 
cially appeals to my mind, as a number of years ago I 
was engaged with similar attempts, of which not the 
beginnings but only a few remnants {Mechanics, pp. 
234-235) are left. I abandoned these attempts, be- 
cause I was convinced that we only apparently evade 
by such expressions references to the fixed stars and 
the angular rotation of the earth. This, in my opin- 
ion, is also true of the forms in which Streintz and 
Lange express the law. 

In point of fact, it was precisely by the considera- 
tion of the fixed stars and the rotation of the earth 
that we arrived at a knowledge of the law of inertia 
as it at present stands, and without these foundations 
we should never have thought of the explanations 
here discussed {Mechanics, 232-233). The considera- 
tion of a small number of isolated points, to the ex- 
clusion of the rest of the world, is in my judgment in- 
admissible {Mechanics, pp. 229-235). 

It is quite questionable, whether a fourth material 
point, left to itself, would, with respect to Lange's 
"inertial system," uniformly describe a straight line, 
if the fixed stars were absent, or not invariable, or 
could not be regarded with sufficient approximation 
as invariable. 

The most natural point of view for the candid in- 
quirer must still be, to regard the law of inertia pri- 
marily as a tolerably accurate approximation, to refer 
it, with respect to space, to the fixed stars, and, with 
respect to time, to the rotation of the earth, and to 
await the correction, or more precise definition, of 



APPEXDIX. 



547 



our knowledge from future experience, as I have ex- 
plained on page 237 of this book. 

I have still to mention the discussions of the law 
of inertia which have appeared since 1889. Reference 
may first be made to the expositions of Karl Pearson 
^Grammar of Science, 1892, page 477), which agree 
with my own, save in terminology. P. and J. Fried- 
lander {Absolute und relative Bewegung, Berlin, 1896) 
have endeavored to determine the question by means 
of an experiment based on the suggestions made by 
me at pages 217-218; I have grave doubts, however, 

ther the experiment will be successful from the 
quantitative side. I can quite freely give my assent 
to the discussions of Johannesson {Das Beharrungs- 
gesetz, Berlin, 1896), although the question remains 
unsettled as to the means by which the motion of a 
body not perceptibly accelerated by other bodies is to 
be determined. For the sake of completeness, the 
predominantly dialectic treatment by M. E. Vicaire, 
Socic'U scientifique de Bruxelles, iSpj, as well as the in- 
vestigations of J. G. MacGregor, Royal Society of Can- 
iSpj, which are only remotely connected with 
the question at issue, remain to be mentioned. I have 
no objections to Budde's conception of space as a sort 
of medium (compare page 230), although I think that 
the properties of this medium should be demonstrable 
physically in some other manner, and that they should 
not be assumed fl// A^. If all apparent actions at a 
distance, all accelerations, turned out to be effected 
through the agency of a medium, then the question 
would appear in a different light, and the solution is 
to be sought perhaps in the view set forth on page 
230. 



548 THE SCIENCE OF MECHANICS. 



XXI. 

(See page 255.) 

Section VIII., "Retrospect of the Development 
of Dynamics," was written in the year 1883. It con- 
tains, especially in paragraph 7, on pages 254 and 
255, an extremely general programme of a future sys- 
tem of mechanics, and it is to be remarked that the 
Mechanics of Hertz, which appeared in the year 1894,* 
marks a distinct advance in the direction indicated. 
It is impossible in the limited space at our disposal 
to give any adequate conception of the copious ma- 
terial contained in this book, and besides it is not our 
purpose to expound new systems of mechanics, but 
merely to trace the development of ideas relating to 
mechanics. Hertz's book must, in fact, be read by 
every one interested in mechanical problems. 

Hertz's criticisms of prior systems of mechanics, 
with which he opens his work, contains some very 
noteworthy epistemological considerations, which from 
our point of view (not to be confounded either with 
the Kantian or with the atomistic mechanical concepts 
of the majority of physicists), stand in need of certain 
modifications. The constructive images f (or better, 
perhaps, the concepts), which we consciously and 
purposely form of objects, are to be so chosen that 
the "consequences which necessarily follow from them 
in thought " agree with the " consequences which nec- 
essarily follow from them in nature." It is demanded 
of these images or concepts that they shall be logically 

*H. Hertz, Die Principien der Mechanik in ?ieuem Zusammenhange dar- 
gestellt. Leipzig, 1894. 

t Hertz uses the term Bild (image or picture) in the sense of the old Eng- 
lish philosophical use of idea, and applies it to systems of ideas or concepts 
relating to any province. 



APPENDIX. 549 

admissible, that is to say, free from all self-contradic- 
tions ; that they shall be correct, that is, shall con- 
form to the relations obtaining between objects; and 
finally that they shall be appropriate, and contain the 
least possible superfluous features. Our concepts, it 
is true, are formed consciously and purposely by us, 
but they are nevertheless not formed altogether arbi- 
trarily, but are the outcome of an endeavor on our 
part to adapt our ideas to our sensuous environment. 
The agreement of the concepts with one another is a 
requirement which is logically necessary, and this 
logical necessity, furthermore, is the only necessity 
that we have knowledge of. The belief in a necessity 
obtaining in nature arises only in cases where our 
concepts are closely enough adapted to nature to 
ensure a correspondence between the logical infer- 
ence and the fact. But the assumption of an adequate 
adaptation of our ideas can be refuted at any moment 
by experience. Hertz's criterion of appropriateness 
coincides with our criterion of economy. 

Hertz's criticism that the Galileo-Newtonian sys- 
tem of mechanics, particularly the notion of force, lacks 
clearness (pages 7, 14, 15) appears to us justified only 
in the case of logically defective expositions, such as 
Hertz doubtless had in mind from his student days. 
He himself partly retracts his criticism in another 
place (pages 9, 47) ; or at any rate, he qualifies it. But 
the logical defects of some individual interpretation 
cannot be imputed to systems as such. To be sure, 
it is not permissible to-day (page 7) "to speak of a 
force acting in one aspect only, or, in the case of cen- 
tripetal force, to take account of the action of inertia 
twice, once as a mass and again as a force." But 
neither is this necessary, since Huygens and Newton 



55Q THE SCIENCE OF MECHANICS. 

were perfectly clear on this point. To characterise 
forces as being frequently "empty-running wheels," 
as being frequently not demonstrable to the senses, 
can scarcely be permissible. In any event, "forces" 
are decidedly in the advantage on this score as com- 
pared with "hidden masses" and "hidden motions." 
In the case of a piece of iron lying at rest on a table, 
both the forces in equilibrium, the weight of the iron 
and the elasticity of the table, are very easily demon- 
strable. 

Neither is the case with energic mechanics so bad 
as Hertz would have it, and as to his criticism against 
the employment of minimum principles, that it in- 
volves the assumption of purpose and presupposes 
tendencies directed to the future, the present work 
shows in another passage quite distinctly that the 
simple import of minimum principles is contained in 
an entirely different property from that of purpose. 
Every system of mechanics contains references to the 
future, since all must employ the concepts of time, 
velocity, etc. 

Nevertheless, though Hertz's criticism of existing 
systems of mechanics cannot be accepted in all their 
severity, his own novel views must be regarded as a 
great step in advance. Hertz, after eliminating the 
concept of force, starts from the concepts of time, 
space, and mass alone, with the idea in view of giv- 
ing expression only to that which can actually be ob- 
served. The sole principle which he employs may 
be conceived as a combination of the law of inertia 
and Gauss's principle of least constraint. Free masses 
move uniformly in straight lines. If they are put in 
connexion in any manner they deviate, in accordance 
with Gauss's principle, as little as possible from this 



APPEXDIX. 551 

motion ; their actual motion is more nearly that of 
free motion than any other conceivable motion. Hertz 
says the masses move as a result of their connexion 
in a straighlcst path. Every deviation of the motion 
of a mass from uniformity and rectilinearity is due, in 
his system, not to a force but to rigid connexion with 
other masses. And where such matters are not vis- 
ible, he conceives hidden masses with hidden motions. 
All physical forces are conceived as the effect of such 
actions. Force, force-function, energy, in his system, 
are secondary and auxiliary concepts only. Let us 
now look at the most important points singly, and 
ask to what extent was the way prepared for them. 
The notion of eliminating force may be reached in 
the following manner. It is part of the general idea 
of the Galileo-Newtonian system of mechanics to 
conceive of all connexions as replaced by forces which 
determine the motions required by the connexions; 
conversely, everything that appears as force may be 
conceived to be due to a connexion. If the first idea 
frequently appears in the older systems, as being his- 
torically simpler and more immediate, in the case of 
Hertz the latter is the more prominent. If we reflect 
that in both cases, whether forces or connexions be 
presupposed, the actual dependence of the motions 
of the masses on one another is given for every in 
stantaneous conformation of the system by linear dif- 
ferential equations between the co ordinates of the 
masses, then the existence of these equations may be 
considered the essential thing, the thing established 
by experience. Physics indeed gradually accustoms 
itself to look upon the description of the facts by dif- 
ferential equations as its proper aim, a point of view 
which was taken also in Chapter V. of the present 



552 THE SCIENCE OF MECHANICS. 

work (1883). But with these the general applicabil- 
ity of Hertz's mathematical formulations is recognised 
without our being obliged to enter upon any further 
interpretation of the forces or connexions. 

Hertz's fundamental law may be described as a 
sort of generalised law of inertia, modified by connex- 
ions of the masses. For the simpler cases, this view 
was a natural one, and doubtless often forced itself 
upon the attention. In fact, the principle of the con- 
servation of the center of gravity and of the conserva- 
tion of areas was actually described in the present 
Work (Chapter III.) as a generalised law of inertia. 
If we reflect that by Gauss's principle the connexion 
of the masses determines a minimum of deviation from 
those motions which it would describe for itself, we 
shall arrive at Hertz's fundamental law the moment we 
consider all the forces as due to the connexions. For 
on severing all connexions, only isolated masses mov- 
ing by the law of inertia are left as ultimate elements. 

Gauss very distinctly asserted that no substantially 
new principle" of mechanics could ever be discovered. 
And Hertz's principle also is only new in form, for it 
is identical with Lagrange's equations. The minimum 
condition which the principle involves does not refer 
to any enigmatic purpose, but its import is the same 
as that of all minimum laws. That alone takes place 
which is dynamically determined (Chapter III.). The 
deviation from the actual motion is dynamically not 
determined; this deviation is not present; the actual 
motion is therefore unique.* 

See Petzoldt's excellent article "Das Gesetz der Eindeutigkeit" [Vier- 
teljahrsschrift fur wissenschaftliche Philosophie, XIX., page 146, especially 
page 186). K. Henke is also mentioned in this article as having approached 
Hertz's view in his tract Ueber die Methode der kleinsten Quadrate (Leipsic, 
1894). 



i 



APPESDIX. 553 

It is hardly necessary to remark that the physical 
side of mechanical problems is not only not disposed 
of, but is not even so much as touched, by the elabo- 
ration of such a formal mathematical system of me- 
chanics. Free masses move uniformly in straight 
lines. Masses having different velocities and direc- 
tions if connected mutually affect each other as to 
velocity, that is, determine in each other accelera- 
tions. These physical experiences enter along with 
purely geometrical and arithmetical theorems into the 
formulation, for which the latter alone would in no 
wise be adequate ; for that which is uniquely deter- 
mined mathematically and geometrically only, is for 
that reason not also uniquely determined mechani- 
cally. But we discussed at considerable length in 
Chapter II., that the physical principles in question 
were not at all self-evident, and that even their exact 
significance was by no means easy to establish. 

In the beautiful ideal form which Hertz has given 
to mechanics, its physical contents have shrunk to 
an apparently almost imperceptible residue. It is 
scarcely to be doubted that Descartes if he lived to- 
day would have seen in Hertz's mechanics, far more 
than in Lagrange's "analytic geometry of four dimen- 
sions," his own ideal. For Descartes, who in his op- 
position to the occult qualities of Scholasticism would 
grant no other properties to matter than extension 
and motion, sought to reduce all mechanics and phys- 
ics to a geometry of motions, on the assumption of a 
motion indestructible at the start. 

It is not difficult to analyse the psychological cir- 
cumstances which led Hertz to his system. After in- 
quirers had succeeded in representing electric and 
magnetic forces that act at a distance as the results 



554 THE SCIENCE OF MECHANICS. 

of motions in a medium, the desire must again have 
awakened to accomplish the same result with respect 
to the forces of gravitation, and if possible for all 
forces whatsoever. The idea was therefore very nat- 
ural to discover whether the concept of force generally 
could not be eliminated. It cannot be denied that 
when we can command all the phenomena taking 
place in a medium, together with the large masses 
contained in it, by means of a single complete pic- 
ture, our concepts are on an entirely different plane 
from what they are when only the relations of these 
isolated masses as regards acceleration are known. 
This will be willingly granted even by those who are 
convinced that the interaction of parts in contact is 
not more intelligible than action at a distance. The 
present tendencies in the development of physics are 
entirely in this direction. 

If we are not content to leave the assumption of 
occult masses and motions in its general form, but 
should endeavor to investigate them singly and in de- 
tail, we should be obliged, at least in the present state 
of our physical knowledge, to resort, even in the sim- 
plest cases, to fantastic and even frequently question- 
able fictions, to which the given accelerations would 
be far preferable. For example, if a mass m is mov- 
ing uniformly in a circle of radius r, with a velocity 
v, which we are accustomed to refer to a centripetal 

force proceeding from the center of the circle, 

r 

we might instead of this conceive the mass to be 
rigidly connected at the distance 2r with one of the 
same size having a contrary velocity. Huygens's cen- 
tripetal impulsion would be another example of a 
force replaced by a connexion. As an ideal program 



APPEXDIX. 



Hertz's mechanics is simpler and more beautiful, but 
for practical purposes our present system of mechan- 
ics is preferable, as Hertz himself (page 47), with his 
characteristic candor, admits.* 



XXII. 
(See page 255.) 

The views put forward in the first two chapters of 
this book were worked out by me a long time ago. 
At the start they were almost without exception coolly 
rejected, and only gradually gained friends. All the 
essential features of my Mechanics I stated originally 
in a brief communication of'five octavo pages entitled 
On the Definition of Mass. These were the theorems 
now given at page 243 of the present book. The 
communication was rejected by Poggendorf's Anna 
len, and it did not appear until a year later (1868), in 
Carl's Repertorium. In a lecture delivered in 1871, I 
outlined my epistemological point of view in natural 
science generally, and with special exactness for phys- 
ics. The concept of cause is replaced there by the 
concept of function; the determining of the depend- 
ence of phenomena on one another, the economic ex- 
position of actual facts, is proclaimed as the object, 
and physical concepts as a means to an end solely. 
I did not care now to impose upon any editor the re- 
sponsibility for the publication of the contents of this 
lecture, and the same was published as a separate 
tract in 1872.! In 1874, when Kirchhoff in his Me- 
chanics came out with his theory of "description " and 

Compare J. Classen, "Die Principien der Mechanik bei Herfz und 
Boltzmann " [Jakrbuch der Hamburgitcken vrittenscka/tlicken Amtabeu, 
XV., p. 1, Hamburg, 1898). 

iErkaUung der Arbeit, Prague, 187a. Sec pf & in ^ M kf' f*'^f 



556 THE SCIENCE OF MECHANICS. 

other doctrines, which were analogous in part only to 
my views, and still aroused the "universal astonish- 
ment" of his colleagues, I became resigned to my 
fate. But the great authority of Kirchhoff gradually 
made itself felt, and the consequence of this also 
doubtless was that on its appearance in 1883 my Me- 
chanics did not evoke so much surprise. In view of 
the great assistance afforded by Kirchhoff, it is alto- 
gether a matter of indifference with me that the pub- 
lic should have regarded, and partly does so still, my 
interpretation of the principles of physics as a contin- 
uation and elaboration of Kirchhoff's views ; whilst in 
fact mine were not only older as to date of publica- 
tion, but also more radical.* 

The agreement with my point of view appears 
upon the whole to be increasing, and gradually to ex- 
tend over more extensive portions of my work. It 
would be more in accord with my aversion for polem- 
ical discussions to wait quietly and merely observe 
what part of the ideas enunciated may be found ac- 
ceptable. But I cannot suffer my readers to remain 
in obscurity with regard to the existing disagreements, 
and I have also to point out to them the way in which 
they can find their intellectual bearings outside of 
this book, quite apart from the fact that esteem for 
my opponents also demands a consideration of their 
criticisms. These opponents are numerous and of all 
kinds: historians, philosophers, metaphysicians, logi- 
cians, educators, mathematicians, and physicists. I 
can make no pretence to any of these qualifications in 
any superior degree. I can only select here the most 
important criticisms, and answer them in the capacity 
of a man who has the liveliest and most ingenuous in- 

*See the preface to the first edition. 



APPENDIX. 



557 



terest in understanding the growth of physical ideas. 
I hope that this will also make it easy for others to 
find their way in this field and to form their own judg- 
ment. 

P. Volkmann in his writings on the epistemology* 
of physics appears as my opponent only in certain 
criticisms on individual points, and particularly by 
his adherence to the old systems and by his predilec- 
tion for them. It is the latter trait, in fact, that sepa- 
rates us ; for otherwise Volkmann's views have much 
affinity with my own. He accepts my adaptation of 
ideas, the principle of economy and of comparison, 
even though his expositions differ from mine in indi- 
vidual features and vary in terminology. I, for my 
part, find in' his writings the important principle of 
isolation and superposition, appropriately emphasised 
and admirably described, and I willingly accept them. 
I am also willing to admit that concepts which at the 
start are not very definite must acquire their "retro- 
active consolidation" by a "circulation of knowl- 
edge," by an "oscillation" of attention. I also agree 
with Volkmann that from this last point of view New- 
ton accomplished in his day nearly the best that it 
was possible to do ; but I cannot agree with Volk- 
mann when he shares the opinion of Thomson and 
Tait, that even in the face of the substantially differ- 
ent epistemological needs of the present day, New- 
ton's achievement is definitive and exemplary. On 
the contrary, it appears to me that if Volkmann's pro- 
cess of "consolidation" be allowed complete sway, it 
must necessarily lead to enunciations not differing in 

* Erkenntnisitheoretisckt Grundtuge dcr Naturaritsensckafl Leipzig, 1896, 
Ucbcr Scojtons Fktlosophia X.ituralis. Konigsberg, 1898. Ein/uhrung in 
dot Stadium dtr theoretitchfn Pkjrtik. Leipsic, ioto. Our references are to 
the last-named work. 



558 THE SCIENCE OF MECHANICS, 

any essential point from my own. I follow with gen- 
uine pleasure the clear and objective discussions of 
G. Heymans.* The differences which I have with 
Hcflerf and PoskeJ relate in the main to individual 
points. So far as principles are concerned, I take 
precisely the same point of view as Petzoldt, and we 
differ only on questions of minor importance. The 
numerous criticisms of others, which either refer to 
the arguments of the writers just mentioned, or are 
supported by analogous grounds, cannot out of regard 
for the reader be treated at length. It will be suffi- 
cient to describe the character of these differences by 
selecting a few individual, but important, points. 

A special difficulty seems to be still found in ac- 
cepting my definition of mass. Streintz (compare p. 
540) has remarked in criticism of it that it is based 
solely upon gravity, although this was expressly ex- 
cluded in my first formulation of the definition (1868). 
Nevertheless, this criticism is again and again put 
forward, and quite recently even by Volkmann {loc. 
cit., p. 18). My definition simply takes note of the 
fact that bodies in mutual relationship, whether it be 
that of action at a distance, so called, or whether rigid 
or elastic connexions be considered, determine in one 
another changes of velocity (accelerations). More than 
this, one does not need to know in order to be able tc 
form a definition with perfect assurance and without 
the fear of building on sand. It is not correct as 

*Die Cesetze unci Elemcnte des wissenschaftlichen Denkcns, II., Leipzig 
1894. 

t Studien zur gegenivdrtigen Philosophie der mathematischcn Mechanik, 
Leipzig, 1900. 

\Vicrteljahrsschrift fur -wissenschaftliche Philosophie, Leipzig, 1884, pag3 

385. 

" Das Gesetz der Eindeutigkeit ( Vierteljahrsschriftfur wissenschaftliche 
Philosophie, XIX., page 146). 



APPENDIX. 559 

Hofler asserts {Joe. cit., p. 77), that this definition 
tacitly assumes one and the same force acting on both 
masses. It does not assume even the notion of force, 
since the latter is built up subsequently upon the no- 
tion of mass, and gives then the principle of action 
and reaction quite independently and without falling 
into Newton's logical error. In this arrangement one 
concept is not misplaced and made to rest on another 
which threatens to give way under it. This is, as I 
take it, the only really serviceable aim of Volkmann's 
"circulation" and "oscillation." After we have de- 
fined mass by means of accelerations, it is not difficult 
to obtain from our definition apparently new variant 
concepts like "capacity for acceleration," "capacity 
for energy of motion" (Hofler, loc. cit., page 70). To 
accomplish anything dynamically with the concept of 
mass, the concept in question must, as I most em- 
phatically insist, be a dynamical concept. Dynamics 
cannot be constructed with quantity of matter by it- 
self, but the same can at most be artificially and arbi- 
trarily attached to it {loc. cit., pages 71, 72). Quantity 
of matter by itself is never mass, neither is it thermal 
capacity, nor heat of combustion, nor nutritive value, 
nor anything of the kind. Neither does "mass" 
play a thermal, but only a dynamical role (compare 
Hofler, loc. cit., pages 71, 72). On the other hand, 
the different physical quantities are proportional to 
one another, and two or three bodies of unit mass 
form, by virtue of the dynamic definition, a body of 
twice or three times the mass, as is analogously the 
case also with thermal capacity by virtue of the ther- 
mal definition. Our instinctive craving for concepts 
involving quantities of things, to which Hofler {loc. 
cit , page 72) is doubtless seeking to give expression, 



560 THE SCIENCE OE MEC11AA 1CS. 

and which amply suffices for every-day purposes, is 
something that no one will think of denying. But 
a scientific concept of "quantity of matter" should 
properly be deduced from the proportionality of the 
single physical quantities mentioned, instead of, con- 
trariwise, building up the concept of mass upon 
"quantity of matter." The measurement of mass by 
means of weight results from my definition quite nat- 
urally, whereas in the ordinary conception the meas- 
urability of quantity of matter by one and the sam< 
dynamic measure is either taken for granted outright, 
or proof must be given beforehand by special experi- 
ments, that equal weights act under all circumstances 
as equal masses. In my opinion, the concept of mass 
has here been subjected to thorough analysis for tht 
first time since Newton. For historians, mathema- 
ticians, and physicists appear to have all treated th< 
question as an easy and almost self evident one. It 
is, on the contrary, of fundamental significance an< 
is deserving of the attention of my opponents. 

Many criticisms have been made of my treatment 
of the law of inertia. I believe I have shown (1868), 
somewhat as Poske has done (1884), that any deduc- 
tion of this law from a general principle, like the law 
of causality, is inadmissible, and this view has now 
won some support (compare Heymans, loc. cit., page 
432). Certainly, a principle that has been universally 
recognised for so short a time only cannot be regarded 
as a priori self-evident. Heymans {Joe. cit., p. 427) cor 
rectly remarks that axiomatic certainty was ascribed 
a few centuries ago to a diametrically opposite form 
of the law. Heymans sees a supra-empirical element 
only in the fact that the law of inertia is referred to 
absolute space, and in the further fact that both in 



APPEXDTX. 561 

the law of inertia and in its ancient diametrically op- 
posite form something constant is assumed in the con- 
dition of the body that is left to itself (Joe. cit., page 
433)- We shall have something to say further on re- 
garding the first point, and as for the latter it is psy- 
chologically intelligible without the aid of metaphys- 
ics, because constant features alone have the power 
to satisfy us either intellectually or practically, 
which is the reason that we are constantly seeking for 
them. Now, looking at the matter from an entirelv 
unprejudiced point of view, the case of these axio- 
matic certainties will be found to be a very peculiar 
one. One will strive in vain with Aristotle to con- 
vince the common man that a stone hurled from the 
hand would be necessarily brought to rest at once after 
its release, were it not for the air which rushed in be- 
hind and forced it forwards. But he would put just 
as little credence in Galileo's theory of infinite uni- 
form motion. On the other hand, Benedetti's theory 
of the gradual diminution of the vis imprcssa, which 
belongs to the period of unprejudiced thought and of 
liberation from ancient preconceptions, will be ac- 
cepted by the common man without contradiction. 
This theory, in fact, is an immediate reflexion of ex- 
perience, while the first-mentioned theories, which 
idealise experience in contrary directions, are a pro- 
duct of technical professional reasoning. They exer- 
cise the illusion of axiomatic certainty only upon the 
mind of the scholar whose entire customary train of 
thought would be thrown out of gear by a disturbance 
of these elements of his thinking. The behavior of 
inquirers toward the law of inertia seems to me from 
a psychological point of view to be adequately ex- 
plained by this circumstance, and I am inclined to 



562 THE SCIENCE OF MECHANICS. 

allow the question of whether the principle is to be 
called an axiom, a postulate, or a maxim, to rest in 
abeyance for the time being. Heymans, Poske, and 
Petzoldt concur in finding an empirical and a supra- 
empirical element in the law of inertia. According to 
Heymans {loc. cit., p. 438) experience simply afforded 
the opportunity for applying an a priori valid prin- 
ciple. Poske thinks that the empirical origin of the 
principle does not exclude its a priori validity {loc. 
cit., pp. 401 and 402). Petzoldt also deduces the law 
of inertia in part only from experience, and regards it 
in its remaining part as given by the law of unique 
determination. I believe I am not at variance with 
Petzoldt in formulating the issue here at stake as fol- 
lows : It first devolves on experience to inform us 
what particular dependence of phenomena on one an- 
other actually exists, what the thing to be determined 
is, and experience alone can instruct us on this 
point. If we are convinced that we have been suffi- 
ciently instructed in this regard, then when adequate 
data are at hand we regard it as unnecessary to keep 
on waiting for further experiences ; the phenomenon 
is determined for us, and since this alone is determi- 
nation, it is uniquely determined. In other words, if 
I have discovered by experience that bodies determine 
accelerations in one another, then in all circumstances 
where such determinative bodies are lacking I shall 
expect with unique determination uniform motion in 
a straight line. The law of inertia thus results imme- 
diately in all its generality, without our being obliged 
to specialise with Petzoldt ; for every deviation from 
uniformity and rectilinearity takes acceleration for 
granted. I believe I am right in saying that the same 
fact is twice formulated in the law of inertia and in 



APPENDIX. 563 

the statement that forces determine accelerations (p. 
143). If this be granted, then an end is also put to 
the discussion as to whether a vicious circle is or is 
not contained in the application of the law of inertia 
(Poske, Hofler). 

Ify inference as to the probable manner in which 
Galileo reached clearness regarding the law of inertia 
was drawn from a passage in his third Dialogue,* 
which was literally transcribed from the Paduan edi- 
tion of 1744, Vol. III., page 124, in my tract on The 
Conservation of Energy (Eng. Trans., in part, in my 
Popular Scientific Lectures, third edition, Chicago, 
The Open Court Publishing Co.). Conceiving a bod 
which is rolling down an inclined plane to be con- 
ducted upon rising inclined planes of varying slopes, 
the slight retardation which it suffers on absolutely 
smooth rising planes of small inclination, and the re- 
tardation zero, or unending uniform motion on a hori- 
zontal plane, must have occurred to him. Wohlwill 
was the first to object to this way of looking at the 
matter (see page 524), and others have since joined 

*" Constat jam, quod mobile ex quiete in A descendens per AB, gradus 
acquirit velocitatis juxta temporis ipsius mcremermim : graoum vero in B 
esse maximum acquisitorum. et suapte natura immutabil ter impressum, 
sublatis scilicet causi> accelerations novae, aut retardationis: accelerations 




inquam. si adhuc super extenso piano ulterius progrederetur : retardationis 
vero. dum super planum acclive BC fie reflexio : in borizontali autem GH 
aeqnabilis motus juxta gradum velocitatis ex A in B acquisitae in infinitum 
jxten.ierpmr." 

It is plain now that a movable body, starting from rest at A and de- 



564 THE SCIENCE OF MECHANICS. 

him. He asserts that uniform motion in a circle and 
horizontal motion still occupied distinct places in Ga- 
lileo's thought, and that Galileo started from the an- 
cient concepts and freed himself only very gradually 
from them. It is not to be denied that the different 
phases in the intellectual development of the great in- 
quirers have much interest for the historian, and some 
one phase may, in its importance in this respect, be 
relegated into the background by the others. One 
must needs be a poor psychologist and have little 
knowledge of oneself not to know how difficult it is 
to liberate oneself from traditional views, and how 
even after that is done the remnants of the old ideas 
still hover in consciousness and are the cause of occa- 
sional backslidings even after the victory has beer 
practically won. Galileo's experience cannot hav< 
been different. But with the physicist it is the in- 
stant in which a new view flashes forth that is of 
greatest interest, and it is this instant for which he 
will always seek. I have sought for it, I believe I 
have found it ; and I am of the opinion that it left its 
unmistakable traces in the passage in question. Poskt 
{/oc. cit., page 393) and Hofler {Joe. eii., pages in, 
ri2) are unable to give their assent to my interpreta- 
tion of this passage, for the reason that Galileo does 
not expressly refer to the limiting case of transition 
from the inclined to the horizontal plane ; although 



scending down the inclined plane AB, acquires a velocity proportional to 
the increment of its time : the velocity possessed at B is the greatest of the 
velocities acquired, and by its nature immutably impressed, provided all 
causes of new acceleration or retardation are taken away : I say accelera- 
tion, having in view its possible further progress along the plane extended ; 
retardation, in view of the possibility of its being reversed and made to 
mount the ascending plane BC. But in the horizontal plane GH its uniform 
motion, with the 1 elocity pcquired in the descent from A to B, will be con- 
tinued nd infi ' 'mti 



APPENDIX. 565 

Poske grants that the consideration of limiting cases 
was frequently employed by Galileo, and although 
Hofler admits having actually tested the educational 
efficacy of this device with students. It would indeed 
be a matter of surprise if Galileo, who may be re- 
garded as the inventor of the principle of continuity, 
should not in his long intellectual career have applied 
the principle to this most important case of all for 
him. It is also to be considered that the passage 
does not form part of the broad and general discus- 
sions of the Italian dialogue, but is tersely couched, in 
the dogmatic form of a result, in Latin. And in this 
way also the "velocity immutably impressed" may 
have crept in.* 

Even granting that Galileo reached his knowledge of the law of inertia 
only gradually, and that it was presented to him merely as an accidental dis- 
covery, nevertheless the following passages which are taken from the Padnan 
edition of 1744 will show that his limitation of the law to horizontal motion 
was justified by the inherent nature of the subject treated ; and the assump- 
tion that Galileo toward the end of his scientific career did not possess a full 
knowledge of the law, can hardly be maintained. 

" Sagr. Ma quando I'artiglieria si piamasse non a perpend icolo. ma in- 
clinata verso qualche parte, qual dovrebbe esser' it moto della pa'la ? and- 
rebbe ella forse, come net l'altro tiro, per la linea perpendicolare, e ritor- 
nando anco poi per 1'istessa ? " 

'Simpl. Questo non farebbe ella, ma uscita del pezzo seguiterebbe il 
suo moto per la linea retta. che continua la dirittura della canna, se non in 
quanto il proprio peso la farebbe declinar da tal dirittura verso terra." 

"Sagr. Talche la dirittura della canna e la regolatrice del moto della 
palla : ne fuori di tal linea si muove, o muoverebbe. se '1 peso proprio non 
la facesse delinare in giii. . . ." Dialogo sopra i du? mass i mi sistetni del 
mondo. 

" Sagr. But jf the gun were not placed in the perpendicular, but were in- 
clined in some direction ; what then would be the motion of the ball ? Would 
it follow, perhaps, as in the other case, the perpendicular, and in returning 
fall also by the same lin'. ? " 

" Simpl. This it will not do, but having left the cannon it will follow its 
own motion in the straight line which is a continuation of the axis of the 
barr-1. save in so far as its own weight shall cause it to deviate from that 
direction toward the earth " 

' Sagr. So that the axis of the barrel is the regulator of the motion of the 
ball: and it ne ther does nor will move outside of that line unless its own 
weight causes it to drop downwards. . . ."* 



566 THE SCIENCE OF MECHANICS. 

The physical instruction which I enjoyed was in 
all probability just as bad and just as dogmatic as it 
was the fortune of my older critics and colleagues to 
enjoy. The principle of inertia was then enunciated 
as a dogma which accorded perfectly with the system. 
I could understand very well that disregard of all ob- 
stacles to motion led to the principle, or that it must 
be discovered, as Apjpelt says, by abstraction ; never- 
theless, it always remained remote and within the 
comprehension of supernatural genius only. And 
where was the guarantee that with the removal of all 
obstacles the diminution of the velocity also ceased? 
Poske (Joe. cit., p. 395) is of the opinion that Galileo, 
to use a phrase which I have repeatedly employed, 
"discerned" or "perceived" the principle immediately. 
But what is this discerning? Enquiring man looks 
here and looks there, and suddenly catches a glimpse 
of something he has been seeking or even of some- 
thing quite unexpected, that rivets his attention. 
Now, I have shown how this "discerning" came 
about and in what it consisted. Galileo runs his eye 
over several different uniformly retarded motions, and 
suddenly picks out from among them a uniform, in- 



"Attendere insuper licet, quod velocitatis gradus, quicunque in mobili 
reperiatur, est in illo suapte natura indelebiliter impressus, diun externae 
causae accelerationis, aut retardationis tollantur, quod in solo horizontali 
piano comingit : nam in plants declivibus adest jam causa accelerationis 
najoris, in acclivibus vero retardationis. Ex quo pariter sequitur, motum in 
horizontali esse quoque aeternum: si enim est aequabilis, non debiliatur, 
aut remittitur, et multo minus tollitur." Discorsi e dimostrazioni matcma- 
tiche. Dialogo terzo. 

" Moreover, it is to be remarked that the degree of velocity a body has is 
indestructibly impressed in it by its own nature, provided external causes of 
acceleration or retardation are wanting, which happens only on horizontal 
planes: for on descending planes there is greater acceleration, and on as- 
cending planes retardation. Whence it follows that motion in a horizontal 
plane is perpetual: for if it remains the same, it is not diminished, or 
abated, much less abolished." 



APPENDIX. 567 

finitely continued motion, of so peculiar a character 
that if it occurred by itself alone it would certainly 
be regarded as something altogether different in kind. 
But a very minute variation of the inclination trans- 
forms this motion into a finite retarded motion, such 
as we have frequently met with in our lives. And 
now, no more difficulty is experienced in recognising 
the identity between all obstacles to motion and re- 
tardation by gravity, wherewith the ideal type of un- 
influenced, infinite, uniform motion is gained. As I 
real this passage of Galileo's while still a young man, 
a new light concerning the necessity of this ideal link 
in our mechanics, entirely different from that of the 
dogmatic exposition, flashed upon me. I believe that 
every one will have the same experience who will ap- 
proach this passage without prior bias. I have not 
the least doubt that Galileo above all others experi- 
enced that light. May my critics see to it how their 
assent also is to be avoided. 

I have now another important point to discuss in 
opposition to C. Neumann,* whose well-known publi- 
cation on this topic preceded minef shortly. I con- 
tended that the direction and velocity which is taken 
into account in the law of inertia had no comprehen- 
sible meaning if the law was referred to "absolute 
space." As a matter of fact, we can metrically deter- 
mine direction and velocity only in a space of which 
the points are marked directly or indirectly by given 
bodies. Neumann's treatise and my own were suc- 
cessful in directing attention anew to this point, which 

*Die Principien der GaliieiScmttm' schen Tkeorie, Leipzig, 1870. 

* Erhaitung der Arbeit \ Prague, 187a. (Translated in part in the article 
on "The Conservation of Energy." Popular Scientific Lectures, third edition, 
Chicago. ,898. <$ ^ ff ^fa * U ft* [> . 4 f . 



568 THE SCIENCE OF MECHANICS. 

had already caused Newton and Euler much intellec- 
tual discomfort; yet nothing more than partial at- 
tempts at solution, like that of Streintz, have resulted. 
I have remained to the present day the only one who 
insists upon referring the law of inertia to the earth, 
and in the case of motions of great spatial and tempo- 
ral extent, to the fixed stars. Any prospect of com- 
ing to an understanding with the great number of my 
critics is, in consideration of the profound differences 
of our points of view, very slight. But so far as I 
have been able to understand the criticisms to which 
my view has been subjected, I shall endeavor to an- 
swer them. 

Hoflems of the opinion that the existence of "ab- 
solute motion" is denied, because it is held to be 
"inconceivable." But it is a fact of "more painstak- 
ing self-observation" that conceptions of absolute 
motion do exist. Conceivability and knowledge of 
absolute motion are not to be confounded. Only the 
latter is wanting here {Joe. cit., pages 120, 164). . . . 
Now, it is precisely with knowledge that the natural 
inquirer is concerned. A thing that is beyond the 
ken of knowledge, a thing that cannot be exhibited to 
the senses, has no meaning in natural science. I have 
not the remotest desire of setting limits to the imagi- 
nation of men, but I have a faint suspicion that the 
persons who imagine they have conceptions of "ab- 
solute motions," in the majority of cases have in mind 
the memory pictures of some actually experienced 
relative motion ; but let that be as it may, for it is in 
any event of no consequence. I maintain even more 
than Hofler, viz., that there exist sensory illusions of 
absolute motions, which can subsequently be repro- 
duced at any time. Every one that has repeated my 



DC, C/t, ffJ^O - l(?4-> 



til'E.\UJX. 569 

experiments on the sensations of movement has ex- 
perience d the full sensory power of such illusions. 
One imagines one is flying off with one's entire en- 
vironment, which remains at relative rest with respect 
to the body ; or that one is rotating in a space that is 
distinguished by nothing that is tangible. But no 
measure can be applied to this space of illusion ; its 
existence cannot be proved to another person, and it 
cannot be employed for the metrical and conceptual 
description of the facts of mechanics; it has nothing 
to do with the space of geometry.* Finally, when 
HQfler {Joe. cit., p. 133) brings forward the argument 
that "in every relative motion one at least of the 
bodies moving with reference to each other must be 
affected with absolute motion," I can only say that 
for the person who considers absolute motion as mean- 
ingless in physics, this argument has no force what- 
ever. But I have no further concern here with philo- 
sophical questions. To go into details as HQfler has 
in some places (loc. cit., pp. 124-126) would serve 
no purpose before an understanding had been reached 
on the main question. ^-/J Hty*} 

Hey mans (Joe. eit., pp. 41 2,-448) remarks that an 
inductive, empirical mechanics could have arisen, but 
that as a matter of fact a different mechanics, based 
on the non-empirical concept of absolute motion, has 
arisen. The fact that the principle of inertia has 
always been suffered to hold for absolute motion 
which is nowhere demonstrable, instead of being re- 



* I flatter myself on being able to resist the temptation to infuse lightness 
into a serious discussion by showing its ridiculous side, but in reflecting on 
these problems I was involuntarily forced to think of the question which a 
very estimable but eccentric man once debated with ine as to whether a yard 
of cloth in one's dreams is as long as a real yard of cloth. -Is the dream-yard 
to be really introduced into mechanics as a standard of measurement ? 



570 THE SCIENCE OF MECHANICS. 

garded as holding good for motion with respect to 
some actually demonstrable system of co-ordinates, is 
a problem which is almost beyond power of solution 
by the empirical theory. Heymans regards this as a 
problem that can have a metaphysical solution only. 
In this I cannot agree with Heymans. He admits 
that relative motions only are given in experience. 
With this admission, as with that of the possibility of 
an empirical mechanics, I am perfectly content. The 
rest, I believe, can be explained simply and without 
the aid of metaphysics. The first dynamic principles 
were unquestionably built up on empirical founda- 
tions. The earth was the body of reference ; the tran- 
sition to the other co-ordinate systems took place 
very gradually. Huygens saw that he could refer the 
motion of impinging bodies just as easily to a boat on 
which they were placed, as to the earth. The devel- 
opment of astronomy preceded that of mechanics con- 
siderably. When motions were observed that were 
at variance with known mechanical laws when re- 
ferred to the earth, it was not necessary immediately 
to abandon these laws again. The fixed stars were 
present and ready to restore harmony as a new sys- 
tem of reference with the least amount of changes in 
the concepts. Think only of the oddities and difficul- 
ties which would have resulted if in a period of great 
mechanical and physical advancement the Ptolemaic 
system had been still in vogue, a thing not at all in- 
conceivable. 

But Newton referred all of mechanics to absolute 
space! Newton is indeed a gigantic personality; little 
worship of authority is needed to succumb to his in- 
fluence. Yet even his achievements are not exempt 
from criticism. It appears to be pretty much one and 



APPENDIX. 571 

the same thing whether we refer the laws of motion to 
absolute space, or enunciate them in a perfectly ab- 
stract form ; that is to say, without specific mention 
of any system of reference. The latter course is un 
precarious and even practical; for in treating special 
cases every student of mechanics looks for some ser 
viceable system of reference. But owing to the fact 
that the first course, wherever there was any real 
issue at stake, was nearly always interpreted as having 
the same meaning as the latter, Newton's error was 
fraught with much less danger than it would other- 
wise have been, and has for that reason maintained 
itself so long. It is psychologically and historically 
intelligible that in an age deficient in epistemological 
critique empirical laws should at times have been 
elaborated to a point where they had no meaning. It 
cannot therefore be deemed advisable to make meta- 
physical problems out of the errors and oversights of 
our scientific forefathers, but it is rather our duty to 
correct them, be they small people or great. I would 
not be understood as saying that this has never hap- 
pened. 

Petzoldt (loc. at., pp. 192 et seq. ), who is in ac- 
cord with me in my rejection of absolute motion, ap- 
peals to a principle of Avenarius,* by a consideration 
of which he proposes to remove the difficulties in- 
volved in the problem of relative motion. I am per- 
fectly familiar with the principle of Avenarius, but I 
cannot understand how all the physical difficulties in- 
volved in the present problem can be avoided by re- 
ferring motions to one's own body. On the contrary, 
in considering physical dependencies abstraction must 

*Der menschlicke Wtltbrgrtjff', Leipzig, 1891, p. 130. 



572 THE SCIENCE OF MECHANICS. 

be made from one's own body, so far as it exercises 
any influence.* 

The most captivating reasons for the assumption 
of absolute motion were given thirty years ago by C. 
Neumann (/oc. cil., p. 27). If a heavenly body be 
conceived rotating about its axis and consequently 
subject to centrifugal forces and therefore" oblate, 
nothing, so far as we can judge, can possibly be 
altered in its condition by the removal of all the re- 
maining heavenly bodies. The body in question will 
continue to rotate and will continue to remain oblate. 
But if the motion be relative only, then the case of 
rotation will not be distinguishable from that of rest. 
All the parts of the heavenly body are at rest with re- 
spect to one another, and the oblateness would neces- 
sarily also disappear with the disappearance of the 
rest of the universe. I have two objections to make 
here. Nothing appears to me to be gained by making 
a meaningless assumption for the purpose of eliminat- 
ing a contradiction. Secondly, the celebrated mathe- 
matician appears to me to have made here too free a 
use of intellectual experiment, the fruitfulness and 
value of which cannot be denied. When experiment- 
r.g in thought, it is permissible to modify unimportant 
circumstances in order to bring out new features in a 
given case; but it is not to be antecedently assumed 
that the universe is without influence on the phenom- 
enon here in question. If it is eliminated and contra- 
dictions still result, certainly this speaks in favor of 
qL the importance of relative motion, which, if it involves 

ii Q ' difficulties, is at least free from contradictions. 

* Analyse tier Empfindungen , zwpite Anflage, Jena. 1900, pp. it, 12. 33, 38, 
208; English translation, Chicago, The Open Court Pub. Co., 1897, pp. 13 et 
seq. 



APPEXDIX. 573 

Yolkmann {Joe. cit., p. 53) advocates an absolute 
orientation by means of the ether. I have already 
spoken on this point (comp. pp. 230, 547), but I am 
extremely curious to know how one ether particle is to 
be distinguished from another. Until some means of 
distinguishing these particles is found, it will be pref- 
erable to abide by the fixed stars, and where these 
forsake us to confess that the true means of orienta- 
tion is still to be found. 

Taking everything together, I can only say that I 
cannot well see what is to be altered in my exposi- 
tions The various points stand in necessary connex- 
ion. After it has been discovered that the behavior 
of bodies toward one another is one in which acceler- 
ations are determined, a discovery which was twice 
formulated by Galileo and Newton, once in a general 
and again in a special form as a law of inertia, it is 
possible to give only one rational definition of mass, 
and that a purely dynamical definition. It is not at 
all, in my judgment, a matter of taste.* The concept 
of force and the principle of action and reaction fol- 
low of themselves. And the elimination of absolute 
motion is equivalent to the elimination of what is 
physically meaningless. 

It would be not only taking a very subjective and 
short-sighted view of science, but it would also be 
foolhardy in the extreme, were I to expect that my 
views in their precise individual form should be in- 
corporated without opposition into the intellectual 
systems of my contemporaries. The history of sci- 
ence teaches that the subjective, scientific philoso- 



* My definition of mass takes a more organic and more natural place i 
Hertz's mechanics than his own. for it contains implicitly the germ of hi- 
"fundamental law " 



574 THE SCIENCE OF MECHANICS. 

phies of individuals are constantly being corrected 
and obscured, and in the philosophy or constructive 
image of the universe which humanity gradually 
adopts, only the very strongest features of the 
thoughts of the greatest men are, after some lapse of 
time, recognisable. It is merely incumbent on the in- 
dividual to outline as distinctly as possible the main 
features of his own view of the world. 

XXIII. 
(See page 273.) 

Although signal individual performances in sci- 
ence cannot be gainsaid to Descartes, as his studies 
on the rainbow and his enunciation of the law of re- 
fraction show, his importance nevertheless is con- 
tained rather in the great general and revolutionary 
ideas which he promulgated in philosophy, mathe- 
matics, and the natural sciences. The maxim of 
doubting everything that has hitherto passed for es- 
tablished truth cannot be rated too high ; although it 
was more observed and exploited by his followers 
than by himself. Analytical geometry with its mod- 
ern methods is the outcome of his idea to dispense 
with the consideration of all the details of geometrical 
figures by the application of algebra, and to reduce 
everything to the consideration of distances. He was 
a pronounced enemy of occult qualities in physics, 
and strove to base all physics on mechanics, which 
he conceived as a pure geometry of motion. He has 
shown by his experiments that he regarded no physi- 
cal problem as insoluble by this method. He took 
too little note of the fact that mechanics is possible 
only on the condition that the positions of the bodies 
are determined in their dependence on one another by 



APPENDIX. 575 

a relation of force, by a function of time; and Leibnitz 
frequently referred to this deficiency. The mechani- 
cal concepts which Descartes developed with scanty 
and vague materials could not possibly pass as copies 
of nature, and were pronounced to be phantasies even 
by Pascal, Huygens, and Leibni/z. It has been re- 
marked, however, in a former place, how strongly 
Descartes's ideas, in spite of these facts, have per- 
sisted to the present day. He also exercises a power- 
ful influence upon physiology by his theory of vision, 
and by his contention that animals were machines, 
a theory which he naturally had not the courage to 
extend to human beings, but by which he anticipated 
the idea of reflex motion (compare Duhem, Devolution 
des theories physiques, Louvain, 1896). 

XXIV. 

(See page 378.; 

To the exposition given on pages 377 and 378, in 
the year 1883, I have the following remarks to add. 
It will be seen that the principle of least action, like 
all other minimum principles in mechanics, is a sim- 
ple expression of the fact that in the instances in 
question precisely so much happens as possibly can 
happen under the circumstances, or as is determined, 
viz., uniquely determined, by them. The deduction 
of cases of equilibrium from unique determination has 
already been discussed, and the same question will be 4 ' / XX 
considered in a later place. /"With respect to dynamic 
questions, the import of the principle of unique de- 
termination has been better and more perspicuously 
elucidated than in my case by J. Petzoldt in a work 
entitled Maxima, Minima und Oekonomie (Altenburg, 
1891). He says {Joe. cit., page 11): "In the case of 



ft ' r- 



576 THE SCIENCE OF MECHANICS. 

all motions, the paths actually traversed admit of be- 
ing interpreted as signal instances chosen from an in- 
finite number of conceivable instances. A